Patchworking

For a set \(\Gamma \subset \R ^2\) and a polynomial \(a(x,y)=\sum _{\omega \in \Bbb Z^2}a_\omega x^{\omega _1}y^{\omega _2}\), denote the polynomial \(\sum _{\omega \in \Gamma \cap \Bbb Z^2}a_\omega x^{\omega _1}y^{\omega _2}\) by \(a^\Gamma \). It is called the \(\GG \)-truncation of \(a\).

For a set \(U\subset \R ^2\) and a real polynomial \(a\) in two variables, denote the curve \(\{(x,y)\in U\,|\,a(x,y)=0\}\) by \(V_U(a)\).

The complement of the coordinate axes in \(\R ^2\), i.e. a set \(\{(x,y)\in \R ^2\,|\, xy\not =0\}\), is denoted This notation is motivated in Section 2.3 below.× ³ by \(\R \R ^2\).

Denote by \(l\) the map \(\R \R ^2\to \R ^2\) defined by formula \(l(x,\) \(y)=(\ln |x|,\ln |y|)\). It is clear that the restriction of \(l\) to each quadrant is a diffeomorphism.

A polynomial in two variables is said to be quasi-homogeneous if its Newton polygon is a segment. The simplest real quasi-homogeneous polynomials are binomials of the form \(\alpha x^p+\beta y^q\) where \(p\) and \(q\) are relatively prime. A curve \(V_{\Bbb {RR}^2}(a)\), where \(a\) is a binomial, is called quasiline. The map \(l\) transforms quasilines to lines. In that way any line with rational slope can be obtained. The image \(l(V_{\Bbb {RR}^2}(a))\) of the quasiline \(V_{\Bbb {RR}^2}(a)\) is orthogonal to the segment \(\Delta (a)\).

It is clear that any real quasi-homogeneous polynomial in 2 variables is decomposable into a product of binomials of the type described above and trinomials without zeros in \(\Bbb {RR}^2\). Thus if \(a\) is a real quasi-homogeneous polynomial then the curve \(V_{\Bbb {RR}^2(a)}\) is decomposable into a union of several quasilines which are transformed by \(l\) to lines orthogonal to \(\Delta (a)\).

A real polynomial \(a\) in two variables is said to be peripherally nondegenerate if for any side \(\Gamma \) of its Newton polygon the curve \(V_{\Bbb {RR}^2}(a^{\Gamma })\) is nonsingular (it is a union of quasilines transformed by \(l\) to parallel lines, so the condition that it is nonsingular means absence of multiple components). Being peripherally nondegenerate is typical in the sense that among polynomials with the same Newton polygons the peripherally nondegenerate ones form nonempty set open in the Zarisky topology.

For a side \(\Gamma \) of a polygon \(\Delta \), denote by \(DC_{\Delta }^-(\Gamma )\) a ray consisting of vectors orthogonal to \(\Gamma \) and directed outside \(\Delta \) with respect to \(\Gamma \) (see Figure 6 and Section 2.2 ).

Figure 6 .

The assertion in the beginning of this Section about behavior of a curve nearby the coordinate axes and at infinity can be made now more precise in the following way.

Theorem generalizing this proposition is formulated in Section 6.3 and proved in Section 6.4 . Here we restrict ourselves to the following elementary example illustrating 1.2.A .

Figure 7 .

Figure 8 .

[b]
Figure 9 .

To any convex closed polygon \(\Delta \subset \R ^2\) with vertices whose coordinates are integers, a real algebraic surface \(\R \Delta \) is associated. This surface is a completion of \(\Bbb {RR}^2\) (\(=(\R \smallsetminus 0)^2\)). The complement \(\R \Delta \smallsetminus \R \R ^2\) consists of lines corresponding to sides of \(\Delta \). From the topological viewpoint \(\R \Delta \) can be obtained from four copies of \(\Delta \) by pairwise gluing of their sides. For a real polynomial \(a\) in two variables we denote the closure of \(V_{\R \R ^2}(a)\) in \(\R \Delta \) by \(V_{\R \Delta }(a)\). Let \(a\) be a real polynomial in two variables which is not quasi-homogeneous. (The latter assumption is not necessary, it is made for the sake of simplicity.) Cut the surface \(\R \Delta (a)\) along lines of \(\R \Delta (a)\smallsetminus \R \R ^2\) (i.e. replace each of these lines by two lines). The result is four copies of \(\Delta (a)\) and a curve lying in them obtained from \(V_{\R \Delta (a)}(a)\). The pair consisting of these four polygons and this curve is a chart of \(a\).

Recall that for \(\Ge ,\delta =\pm 1\) we denote the reflection \(\RR ^2\to \RR ^2:(x,y)\mapsto (\Ge x,\delta y)\) by \(S_{\Ge ,\delta }\). For a set \(A\subset \R ^2\) we denote \(S_{\Ge ,\delta }(A)\) by \(A_{\Ge ,\delta }\) (see Figure 2 ). Denote a quadrant \(\{(x,y)\in \R ^2\,|\,\Ge x>0, \delta y>0\}\) by \(Q_{\Ge ,\delta }\).

Now define the charts for two classes of real polynomials separately.

First, consider the case of quasi-homogeneous polynomials. Let \(a\) be a quasi-homogeneous polynomial defining a nonsingular curve \(V_{\R \R ^2}(a)\). Let \((w_1,w_2)\) be a vector orthogonal to \(\Delta = \Delta (a)\) with integer relatively prime coordinates. It is clear that in this case \(V_{\R ^2}(a)\) is invariant under \(S_{(-1)^{w_1},(-1)^{w_2}}\). A pair \((\Delta _*,\) \(\upsilon )\) consisting of \(\Delta _*\) and a finite set \(\upsilon \subset \Delta _*\) is called a chart of \(a\), if the number of points of \(\upsilon \cap \Delta _{\Ge ,\delta }\) is equal to the number of components of \(V_{Q_{\Ge ,\delta }}(a)\) and \(\upsilon \) is invariant under \(S_{(-1)^{w_1},(-1)^{w_2}}\) (remind that \(V_{\R ^2}(a)\) is invariant under the same reflection).

Example 1.3.A.

In Figure 10 it is shown a curve \(V_{\R ^2}(a)\) with \(a(x,y) = 2x^6y - x^4y^2 - 2x^2y^3 + y^4 = (x^2 - y)(x^2 + y)(2x^2 - y)y\), and a chart of \(a\).

Figure 10 .

Now consider the case of peripherally nondegenerate polynomials with Newton polygons having nonempty interiors. Let \(\Delta \), \(\Gamma _1,\dots ,\Gamma _n\) and \(a\) be as in 1.2.A . Then, as it follows from 1.2.A , there exist a disk \(D\subset \R ^2\) with center at the origin and neighborhoods \(D_1,\dots ,D_n\) of rays \(DC^-_\Delta (\Gamma _1),\dots ,DC^-_\Delta (\Gamma _n)\) such that the curve \(V_{\R \R ^2}(a)\) lies in \(l^{-1}(D\cup D_1\cup \dots \cup D_n)\) and for \(i = 1,\dots ,n\) the curve \(V_{l^{-1}(D_i\smallsetminus D)}(a)\) is approximated by \(V_{l^{-1}(D_i\smallsetminus D)}(a^{\Gamma _i})\) and can be contracted (in itself) to \(V_{l^{-1}(D_i\cap \p D)}(a)\).

A pair \((\Delta _*,\) \(\upsilon )\) consisting of \(\Delta _*\) and a curve \(\upsilon \subset \Delta _*\) is called a chart of \(a\) if

1. for \(i = 1,\dots ,n\) the pair \((\Gamma _{i*},\) \(\Gamma _{i*}\cap \upsilon )\) is a chart of \(a^{\Gamma _i}\) and
2. for \(\Ge \), \(\delta = \pm 1\) there exists a homeomorphism \(h_{\Ge ,\delta }:D\to \Delta \) such that \(\upsilon \cap \Delta _{\Ge , \delta } = S_{\Ge ,\delta }\circ h_{\Ge ,\delta }\circ l(V_{l^{-1}(D)\cap Q_{\Ge ,\delta }}(a))\) and \(h_{\Ge ,\delta } (\p D\cap D_i)\subset \Gamma _i\) for \(i = 1,\dots ,n\).

It follows from 1.2.A that any peripherally nondegenerate real polynomial \(a\) with \(\Int \Delta (a)\ne \empt \) has a chart. It is easy to see that the chart is unique up to a homeomorphism \(\Delta _*\to \Delta _*\) preserving the polygons \(\Delta _{\Ge ,\delta }\), their sides and their vertices.

Figure 11 .

Proof.

Consider the restriction of \(a\) to the quadrant \(Q_{\Ge ,\Gd }\). If all signs \(\Gs _{\Ge i_k,\Gd j_k}\) are the same, then \(a{Q_{\Ge ,\Gd }}\) is a sum of three monomials taking values of the same sign on \(Q_{\Ge ,\Gd }\). In this case \(V_{Q_{\Ge ,\Gd }}(a)\) is empty. Otherwise, consider the side \(\GG \) of the triangle \(\GD \) on whose end points the signs coincide. Take a vector \((w_1,w_2)\) orthogonal to \(\GG \). Consider the curve defined by parametric equation \(t\mapsto (x_0t^{w_1},y_0t^{w_2})\). It is easy to see that the ratio of the monomials corresponding to the end points of \(\GG \) does not change along this curve, and hence the sum of them is monotone. The ratio of each of these two monomials with the third one changes from \(0\) to \(-\infty \) monotonically. Therefore the trinomial divided by the monomial which does not sit on \(\GG \) changes from \(-\infty \) to \(1\) continuously and monotonically. Therefore it takes the zero value once. Curves \(t\mapsto (x_0t^{w_1},y_0t^{w_2})\) are disjoint and fill \(Q_{\Ge ,\Gd }\). Therefore, the curve \(V_{Q_{\Ge ,\Gd }}(a)\) is isotopic to the preimage under \(S_{\Ge ,\Gd }\circ h_{\Ge ,\Gd }\circ l\) of the midline of the triangle \(\GD _{\Ge ,\Gd }\) separating the vertices with opposite signs.

The part of this proposition concerning to \(V_{\R \R ^2}(a)\) follows from 1.2.A . See below Sections 2 and 3 about toric extensions of \(\R \R ^2\) and closures of \(V_{\R \R ^2}(a)\) in them. In the next Subsection algorithms recovering the topology of closures of \(V_{\R \R ^2}(a)\) in \(\R ^2\) and \(\rpp \) from a chart of \(a\) are described.

Let \(\nu :\Delta \to \R \) be a convex function such that:

1. restrictions \(\nu |_{\Delta (a_i)}\) are linear;
2. if the restriction of \(\nu \) to an open set is linear then the set is contained in one of \(\Delta (a_i)\);
3. \(\nu (\Delta \cap \Bbb Z^2)\subset \Bbb Z\).

If \(a(x,y)=\sum _{\omega \in \Bbb Z^2}a_\omega x^{\omega _1}y^{\omega _2}\) then we put

\[b_t(x,y)=\sum _{\omega \in \Bbb Z^2}a_\omega x^{\omega _1}y^{\omega _2}t^{\nu (\omega _1,\omega _2)}\]

and say that polynomials \(b_t\) are obtained by patchworking \(a_1,\dots ,a_s\) by \(\nu \).

By 1.3.C , Theorem 1.7.A generalizes Theorem 1.1.D . Theorem generalizing Theorem 1.7.A is proven in Section 4.3 . Here we restrict ourselves to several examples.

In the next Section there are a number of considerably more complicated examples demonstrating efficiency of Theorem 1.7.A in the topology of real algebraic curves.

Figure 16 .

Gudkov proposed twice — in [Gud73] and [Gud71] — simplified proofs of realizability of the third isotopy type. His constructions, however, are essentially more complicated than the construction described below, which is based on 1.7.A and besides gives rise to realization of the other two types, and, after a small modification, realization of almost all isotopy types of nonsingular plane projective real algebraic curves of degree 6 (see [Vir89] ).

Figure 17 .

Construction In Figure 17 two curves of degree 6 are shown. Each of them has one singular point at which three nonsingular branches are second order tangent to each other (i.e. this singularity belongs to type \(J_{10}\) in the Arnold classification [AVGZ82] ). The curves of Figure 17 (a) and (b) are easily constructed by the Hilbert method [Hil91] , see in [Vir89] , Section 4.2.

Choosing in the projective plane various affine coordinate systems, one obtains various polynomials defining these curves. In Figures 18 and 19 charts of four polynomials appeared in this way are shown. In Figure 20 the results of patchworking charts of Figures 18 and 19 are shown. All constructions can be done in such a way that Theorem 1.7.A (see [Vir89] , Section 4.2) may be applied to the corresponding polynomials. It ensures existence of polynomials with charts shown in Figure 20 .

	Figure 18 .
Figure 19 .	Figure 20 .

How do the domains containing the pieces of \(V_{\R \R ^2}(b_t)\) homeomorphic to \(V_{\R \R ^2}(a_1)\), …, \(V_{\R \R ^2}(a_s)\) behave when \(t\) approaches zero? They are moving to the coordinate axes and infinity. The closer \(t\) to zero, the more place is occupied by the domain, where \(V_{\R \R ^2}(b_t)\) is organized as \(V_{\R \R ^2}(a_1)\) and is approximated by it (cf. Section 6.7 ).

It is curious that the family \(b_t\) can be changed by a simple geometric transformation in such a way that the role of \(a_1\) passes to any one of \(a_2,\dots ,a_s\) or even to \(a_k^\Gamma \), where \(\Gamma \) is a side of \(\Delta (a_k)\), \(k=1, \dots , s\). Indeed, let \(\lambda :\R ^2\to \R \) be a linear function, \(\lambda (x,y)=\alpha x+\beta y+\gamma \). Let \(\nu '=\nu -\lambda \). Denote by \(b'_t\) the result of patchworking \(a_1,\dots ,a_s\) by \(\nu '\). Denote by \(qh_{(a,b),t}\) the linear transformation \(\R \R ^2\to \R \R ^2:(x,y)\mapsto (xt^a,yt^b)\). Then

\[V_{\R \R ^2}(b'_t)=V_{\R \R ^2}(b_t\circ qh_ {(-\alpha ,-\beta ),t})=qh_{(\alpha ,\beta ),t}V_{\R \R ^2}(b_t).\]

Indeed,

\[\begin {aligned} b'_t(x,y) =&\sum a_\omega x^{\omega _1}y^{\omega _2}t^{\nu (\omega _1,\omega _2)-\alpha \omega _1- \beta \omega _2 -\gamma } \\ &= t^{-\gamma }\sum a_\omega (xt^{-\alpha })^{\omega _1} (yt^{-\beta })^{\omega _2}t^{\nu (\omega _1,\omega _2)} \\ &= t^{-\gamma }b_t(xt^{-\alpha },yt^{-\beta }) \\ &= t^{-\gamma }b_t\circ qh_{(-\alpha ,-\beta ),t}(x,y). \end {aligned}\]

Thus the curves \(V_{\R \R ^2}(b'_t)\) and \(V_{\R \R ^2}(b_t)\) are transformed to each other by a linear transformation. However the polynomial \(b'_t\) does not tend to \(a_1\) as \(t\to 0\). For example, if \(\lambda |_{\Delta (a_k)}=\nu |_{\Delta (a_k)}\) then \(\nu '|_{\Delta (a_k)}=0\) and \(b'_t\to a_k\). In this case as \(t\to 0\), the domains containing parts of \(V_{\R \R ^2}(b'_t)\), which are homeomorphic to \(V_{\R \R ^2}(a_i)\), with \(i\ne k\), run away and the domain in which \(V_{\R \R ^2}(b'_t)\) looks like \(V_{\R \R ^2}(a_k)\) occupies more and more place. If the set, where \(\nu \) coincides with \(\lambda \) (or differs from \(\lambda \) by a constant), is a side \(\Gamma \) of \(\Delta (a_k)\), then the curve \(V_{\R \R ^2}(b'_t)\) turns to \(V_{\R \R ^2}(a_k^\Gamma )\) (i.e. collection of quasilines) as \(t\to 0\) similarly.

The whole picture of evolution of \(V_{\R \R ^2}(b_t)\) when \(t\to 0\) is the following. The fragments which look as \(V_{\R \R ^2}(a_i)\) with \(i=1,\dots ,s\) become more and more explicit, but these fragments are not staying. Each of them is moving away from the others. The only fragment that is growing without moving corresponds to the set where \(\nu \) is constant. The other fragments are moving away from it. From the metric viewpoint some of them (namely, ones going to the origin and axes) are contracting, while the others are growing. But in the logarithmic coordinates, i.e. being transformed by \(l:(x,y)\mapsto (\ln |x|,\ln |y|)\), all the fragments are growing (see Section 6.7 ). Changing \(\nu \) we are applying linear transformation, which distinguishes one fragment and casts away the others. The transformation turns our attention to a new piece of the curve. It is as if we would transfer a magnifying lens from one fragment of the curve to another. Naturally, under such a magnification the other fragments disappear at the moment \(t=0\).

Figure 21 .

Figure 22 .

Curves of degree 6 with eleven components of all three isotopy types can be obtained from this curve by small perturbations of the type under consideration (cf. Section 1.8 ). Moreover, as it is proven in [Vir89] , Section 5.1, nonsingular curves of degree 6 of almost all isotopy types can be obtained.

Let \(f\) be included into a continuous family \(f_t\) of polynomials in two variables: \(f=f_0\). Such a family is called a perturbation of \(f\). We shall be interested mainly in perturbations for which curves \(V_{\R ^2}(f_t)\) have no singular points in \(B\) when \(t\) is in some segment of type \((0,\Ge ]\). One says about such a perturbation that it evolves the singularity of \(V_{\Bbb R^2}(f_t)\) at zero. If perturbation \(f_t\) evolves the singularity of \(V_{\R ^2}(f)\) at zero then one can find \(t_0>0\) such that for \(t\in (0,t_0]\) the curve \(V_{\R ^2}(f_t)\) has no singularities in \(B\) and, moreover, is transversal to \(\p B\). Obviously, there exists an isotopy \(h_t:B\to B\) with \(t_0\in (0,t_0]\) such that \(h_{t_0}=\id \) and \(h_t(V_B(f_0))=V_B(f_t)\), so all pairs \((B, V_B(f_t))\) with \(t\in (0,t_0]\) are homeomorphic to each other. A family \((B,V_{\R ^2}(f_t))\) of pairs with \(t\in (0,t_0]\) is called an evolving of singularity of \(V_{\R ^2}(f)\) at zero, or an evolving of germ of \(V_{\R ^2}(f)\).

Denote by \(\Gamma _1, \dots ,\Gamma _n\) the sides of Newton polygon \(\Delta (f)\) of the polynomial \(f\), faced to the origin. Their union \(\Gamma (f)=\bigcup _{i=1}^n\Gamma _i\) is called the Newton diagram of \(f\).

Suppose the curves \(V_{\R \R ^2}(f^{\Gamma _i})\) with \(i=1,\dots ,n\) are nonsingular. Then, according to Newton [New67] , the curve \(V_{\R ^2}(f)\) is approximated by the union of \(\Cl V_{\R \R ^2}(f^{\Gamma _i})\) with \(i=1,\dots ,n\) in a sufficiently small neighborhood of the origin. (This is a local version of Theorem 1.2.A ; it is, as well as 1.2.A , a corollary of Theorem 6.3.A .) Disk \(B\) can be taken so small that \(V_{\p B}(f)\) is close to \(\p B\cap V_{\R \R ^2}(f^{\Gamma _i})\), so the number and disposition of these points are defined by charts \((\Gamma _{i*},\,\upsilon _i)\) of \(f^\Gamma _i\). The union \((\Gamma (f)_*,\,\upsilon )= (\bigcup _{i=1}^n\Gamma _{i*},\,\bigcup _{i=1}^n\upsilon _i)\) of these charts is called a chart of germ of \(V_{\R ^2}(f)\) at zero. It is a pair consisting of a simple closed polygon \(\Gamma (f_*)\), which is symmetric with respect to the axes and encloses the origin, and finite set \(\upsilon \) lying on it. There is a natural bijection of this set to \(V_{\p B}(f)\), which is extendable to a homeomorphism \((\Gamma (f)_*,\) \(\upsilon )\to (\p B,V_{\p B}(f))\). Denote this homeomorphism by \(g\).

Let \(f_t\) be a perturbation of \(f\), which evolves the singularity at the origin. Let \(B\), \(t_0\) and \(h_t\) be as above. It is not difficult to choose an isotopy \(h_t:B\to B\), \(t\in (0,t_0]\) such that its restriction to \(\p B\) can be extended to an isotopy \(h'_t:\p B\to \p B\) with \(t\in [0,t_0]\) and \(h'_0(V_{\p B}(f_{t_0}))=V_{\p B}(f)\). A pair \((\Pi ,\tau )\) consisting of the polygon \(\Pi \) bounded by \(\Gamma (f)_*\) and an 1-dimensional subvariety \(\tau \) of \(\Pi \) is called a chart of evolving \((B,V_B(f_t))\), \(t\in [0,t_0]\) if there exists a homeomorphism \(\Pi \to B\), mapping \(\tau \) to \(V_{\p B}(f_{t_0})_*\), whose restriction \(\p \Pi \to \p \Pi \) is the composition \(\Gamma (f)_*@>g>>\p B@>{h'_0}>>\p B\). It is clear that the boundary \((\p \Pi ,\p \tau )\) of a chart of germ's evolving is a chart of the germ. Also it is clear that if polynomial \(f\) is completely nondegenerate and polygons \(\Delta (f_t)\) are obtained from \(\Delta (f)\) by adjoining the region restricted by the axes and \(\Pi (f)\), then charts of \(f_t\) with \(t\in (0,t_0]\) can be obtained by patchworking a chart of \(f\) and chart of evolving \((B,V_B(f_t))\), \(t\in [0,t_0]\).

The patchworking construction for polynomials gives a wide class of evolvings whose charts can be created by the modification of Theorem 1.7.A formulated below.

Let \(a_1, \dots , a_s\) be completely nondegenerate polynomials in two variables with \(\Int \Delta (a_i)\cap \Int \Delta (a_j)=\varnothing \) and \(a_i^{\Delta (a_i)\cap \Delta (a_j)}=a_j^{\Delta (a_i)\cap \Delta (a_j)}\) for \(i\ne j\). Let \(\bigcup _{i=1}^s\Delta (a_i)\) be a polygon bounded by the axes and \(\Gamma (f)\). Let \(a_i^{\Delta (a_i)\cap \Delta (f)}=f^{\Delta (a_i)\cap \Delta (f)}\) for \(i=a, \dots , s\). Let \(\nu :\R ^2\to \R \) be a nonnegative convex function which is equal to zero on \(\Delta (f)\) and whose restriction on \(\bigcup _{i=1}^s\Delta (a_i)\) satisfies the conditions 1.6 , 1.6 and 1.6 of Section 1.6 with respect to \(a_1, \dots ,a_s\). Then a result \(f_t\) of patchworking \(f\), \(a_1, \dots , a_s\) by \(\nu \) is a perturbation of \(f\).

Theorem 1.7.A cannot be applied in this situation because the polynomial \(f\) is not supposed to be completely nondegenerate. This weakening of assumption implies a weakening of conclusion.

An evolving of a germ, constructed along the scheme above, is called a patchwork evolving.

If \(\Gamma (f)\) consists of one segment and the curve \(V_{\R \R ^2}(f^{\Gamma (f)})\) is nonsingular then the germ of \(V_{\R ^2}(f)\) at zero is said to be semi-quasi-homogeneous. In this case for construction of evolving of the germ of \(V_{\R ^2}(f)\) according the scheme above we need only one polynomial; by 1.11.A , its chart is a chart of evolving. In this case geometric structure of \(V_B(f_t)\) is especially simple, too: the curve \(V_B(f_t)\) is approximated by \(qh_{w,t}(V_{\R ^2}(a_1))\), where \(w\) is a vector orthogonal to \(\Gamma (f)\), that is by the curve \(V_{\RR ^2}(a_1)\) contracted by the quasihomothety \(qh_{w,t}\). Such evolvings were described in my paper [Vir80] . It is clear that any patchwork evolving of semi-quasi-homogeneous germ can be replaced, without changing its topological models, by a patchwork evolving, in which only one polynomial is involved (i.e. \(s=1\)).

For \(\Go =(\Go _1,\dots ,\Go _n)\in \Z ^n\) and ordered collection \(x\) of variables \(x_1, \dots ,x_n\) the product \(x_1^{\Go _1}\dots x_n^{\Go _n}\) is denoted by \(x^\Go \). A linear combination of products of this sort with coefficients from \(K\) is called a Laurent polynomial or, briefly, L-polynomial over \(K\). Laurent polynomials over \(K\) in \(n\) variables form a ring \(K[x_1,x_1^{-1},\dots ,x_n,x_n^{-1}]\) naturally isomorphic to the ring of regular functions of the variety \((K\sminus 0)^n\).

Below this variety, side by side with the affine space \(K^n\) and the projective space \(KP^n\), is one of the main places of action. It is an algebraic torus over \(K\). Denote it by \(K\R ^n\).

Denote by \(l\) the map \(K\R ^n\to \R ^n\) defined by formula \(l(x_1,\dots ,x_n)=\) \((\ln |x_1|,\) \(\dots ,\) \(\ln |x_n|)\).

Put \(U_K=\{x\in K\,|\,|x|=1\}\), so \(U_{\R }=S^0\) and \(U_{\C }=S^1\). Denote by \(ar\) the map \(K\R ^n\to U_K^n\) (\(=U_K\times \dots \times U_K)\) defined by \(ar(x_1,\dots ,x_n)=(\dfrac {x_1}{|x_1|},\dots ,\dfrac {x_n}{|x_n|})\).

Denote by \(la\) the map

\[x\mapsto (l(x),ar(x)):K\R ^n\to \R ^n\times U_K^n.\]

It is clear that this is a diffeomorphism.

\(K\R ^n\) is a group with respect to the coordinate-wise multiplication, and \(l\), \(ar\), \(la\) are group homomorphisms; \(la\) is an isomorphism of \(K\R ^n\) to the direct product of (additive) group \(\R ^n\) and (multiplicative) group \(U_K^n\).

Being Abelian group, \(K\R ^n\) acts on itself by translations. Let us fix notations for some of the translations involved into this action.

For \(w\in \R ^n\) and \(t>0\) denote by \(qh_{w,t}\) and call a quasi-homothety with weights \(w=(w_1,\dots ,w_n)\) and coefficient \(t\) the transformation \(K\R ^n\to K\R ^n\) defined by formula \(qh_{w,t}(x_1,\dots ,x_n)=(t^{w_1}x_1,\dots ,t^{w_n}x_n)\), i.e. the translation by \((t^{w_1},\dots ,t^{w_n})\). If \(w=(1,\dots ,1)\) then it is the usual homothety with coefficient \(t\). It is clear that \(qh_{w,t}=qh_{\Gl ^{-1}w,t}\) for \(\Gl >0\). Denote by \(qh_w\) a quasi-homothety \(qh_{w,e}\), where \(e\) is the base of natural logarithms. It is clear, \(qh_{w,t}=qh_{(\ln t)w}\).

For \(w=(w_1,\dots ,w_n)\in U_K^n\) denote by \(S_w\) the translation \(K\R ^n\to K\R ^n\) defined by formula

\[S_w(x_1,\dots ,x_n)=(w_1x_1,\dots ,w_nx_n),\]

i. e. the translation by \(w\).

For \(w\in \R ^n\) denote by \(T_w\) the translation \(x\mapsto x+w:\R ^n\to \R ^n\) by the vector \(w\).

In particular, \(la\circ qh_w\circ la^{-1}=T_w\times \id \).

A hypersurface of \(K\R ^n\) defined by \(a(x)=0\), where \(a\) is a Laurent polynomial over \(K\) in \(n\) variables is denoted by \(V_{K\R ^n}(a)\).

If \(a(x)=\sum _{\Go \in \Z ^n}a_\Go x^\Go \) is a Laurent polynomial, then by its Newton polyhedron \(\GD (a)\) is the convex hull of \(\{\Go \in \R ^n\,|\, a_\Go \ne 0\}\).

Proof.

Proposition 2.1.B is equivalent, as it follows from 2.1.A , to the assertion that under hypothesis of 2.1.B the set \(la(V_{K\R ^n}(a))\) contains together with each point \((x,y)\in \R ^n\times U_K^n\) all points \((x',y)\in \R ^n\times U_K^n\) with \(x'-x\perp \GG \). In other words, in the case \(\GD (a)\subset \GG \) the intersections of \(la(V_{K\R ^n}(a))\) with fibers \(\R ^n\times y\) are cylinders, whose generators are affine spaces of dimension \(n-\dim \GG \) orthogonal to \(\GG \).

The following proposition can be proven similarly to 2.1.B .

In other words, under the hypothesis of 2.1.B the hypersurface \(V_{K\R ^n}(a)\) contains together with each its point \((x_1,\dots ,x_n)\):

1. points \(((-1)^{w_1}x_1,\dots ,(-1)^{w_n}x_n)\) with \(w\in \Z ^n\), \(w\perp \GG \), if \(K=\R \),
2. points \((e^{iw_1}x_1,\dots ,e^{iw_n}x_n)\) with \(w\in \Bbb R^n\), \(w\perp \GG \), if \(K=\C \).

The set of faces of a polyhedron \(\GD \) is denoted by \(\mathcal {G}(\GD )\), the set of its \(k\)-dimensional faces by \(\mathcal {G}_k(\GD )\), the set of all its proper faces by \(\mathcal {G}'(\GD )\).

By a halfspace of vector space \(V\) we will mean the preimage of the closed halfline \(\R _+(=\{x\in \R :x\ge 0\})\) under a non-zero linear functional \(V\to \R \) (so the boundary hyperplane of a halfspace passes necessarily through the origin). By a cone it is called an intersection of a finite collection of halfspaces of \(\R ^n\). A cone is a polyhedron (not necessarily integer), hence all notions and notations concerning polyhedra are applicable to cones.

The minimal face of a cone is the maximal vector subspace contained in the cone. It is called a ridge of the cone.

For \(v_1,\dots ,v_k\in \R ^n\) denote by \(\langle v_1,\dots ,v_k\rangle \) the minimal cone containing \(v_1\), …, \(v_k\); it is called the cone generated by \(v_1,\dots ,v_k\). A cone is said to be simplicial if it is generated by a collection of linear independent vectors, and simple if it is generated by a collection of integer vectors, which is a basis of the free Abelian group of integer vectors lying in the minimal vector space which contains the cone.

Let \(\GD \subset \R ^n\) be a polyhedron and \(\GG \) its face. Denote by \(C_\GD (\GG )\) the cone \(\bigcup _{r\in \R _+}r\cdot (\GD -y)\), where \(y\) is a point of \(\GG \sminus \partial \GG \). The cone \(C_\GD (\GD )\) is clearly the vector subspace of \(\R ^n\) which corresponds to the minimal affine subspace containing \(\GD \). The cone \(C_\GG (\GG )\) is the ridge of \(C_\GD (\GG )\). If \(\GG \) is a face of \(\GD \) with \(\dim \GG =\dim \GD -1\), then \(C_\GD (\GG )\) is a halfspace of \(C_\GD (\GD )\) with boundary parallel to \(\GG \).

For cone \(C\subset \R ^n\) we put

\[D^+C=\{x\in \R ^n\,|\,\forall a\in C \quad ax\ge 0\},\]

\[D^-C=\{x\in \R ^n\,|\,\forall a\in C \quad ax\le 0\}.\]

These are cones, which are said to be dual to \(C\). The cones \(D^+C\) and \(D^-C\) are symmetric to each other with respect to \(0\). The cone \(D^-C\) permits also the following more geometric description. Each hyperplane of support of \(C\) defines a ray consisting of vectors orthogonal to this hyperplane and directed to that of two open halfspaces bounded by it, which does not intersect \(C\). The union of all such rays is \(D^-C\).

It is clear that \(D^+D^+C=C=D^-D^-C\). If \(v_1,\dots ,v_n\) is a basis of \(\R ^n\), then the cone \(D^+\langle v_1,\dots ,v_n\rangle \) is generated by dual basis \(v_1^*,\dots ,v_n^*\) (which is defined by conditions \(v_i\cdot v_j*=\GD _{ij}\)).

An affine toric variety \(K\GD \) is the affine scheme \(\Spec K[\GD \cap \Z ^n]\). Its less invariant, but more elementary definition looks as follows. Let

\[\{\alpha _1,\dots ,\alpha _p\,|\,\sum _{i=1}^p u_{1,i}\alpha _i= \sum _{i=1}^p v_{i,1}\alpha _i,\dots ,\sum _{i=1}^p u_{p-n,i}\alpha _i= \sum _{i=1}^p v_{p-n,i}\alpha _i\}\]

be a presentation of \(\GD \cap \Z ^n\) by generators and relations (here \(u_{ij}\) and \(v_{ij}\) are nonnegative); then the variety \(K\GD \) is isomorphic to the affine subvariety of \(K^p\) defined by the system

\[\begin {cases} \begin {aligned} y_1^{u_{11}}\dots y_p^{u_{1p}}&=y_1^{v_{11}}\dots y_p^{v_{1p}}\\ \dots \\ y_1^{u_{p-n,1}}\dots y_p^{u_{p-n,p}}&=y_1^{v_{p-n,1}}\dots y_p^{v_{p-n,p}}. \end {aligned} \end {cases} \]

For example, if \(\GD =\R ^n\), then \(K\GD =\Spec K[x_1,x_1^{-1},\dots ,x_n,x_n^{-1}]\) can be presented as the subvariety of \(K^{2n}\) defined by the system

\[ \begin {cases}\begin {aligned} y_1y_{n+1}=1\\ \dots \\y_ny_{2n}=1\end {aligned} \end {cases} \]

Projection \(K^{2n}\to K^n\) induces an isomorphism of this subvariety to \((K\sminus 0)^n=K\R ^n\). This explains the notation \(K\R ^n\) introduced above.

If \(\GD \) is the positive orthant \(A^n=\{x\in \R ^n\,|\, x_1\ge 0,\dots ,x_n\ge 0\}\), then \(K\GD \) is isomorphic to the affine space \(K^n\). The same takes place for any simple cone. If cone is not simple, then corresponding toric variety is necessarily singular. For example, the angle shown in Figure 23 corresponds to the cone defined in \(K^3\) by \(xy=z^2\).

Figure 23 .

\[\inj ^*:\Spec K[\GD _2\cap \Z ^n]\to \Spec K[\GD _1\cap \Z ^n],\]

i.e. a regular map \(\inj ^*:K\GD _2\to K\GD _1\). The latter can be described in terms of subvarieties of affine spaces in the following way. The formulas, defining coordinates of point \(\inj ^*(y)\) as functions of coordinates of \(y\), are the multiplicative versions of formulas, defining generators of semigroup \(\GD _1\cap \Z ^n\) as linear combinations of generators of the ambient semigroup \(\GD _2\cap \Z ^n\).

In particular, for any \(\GD \) there is a regular map of \(KC_\GD (\GD )\cong K\R ^{\dim \GD }\) to \(K\GD \). It is not difficult to prove that it is an open embedding with dense image, thus \(K\GD \) can be considered as a completion of \(K\R ^{\dim \GD }\).

An action of algebraic torus \(KC_\GD (\GD )\) in itself by translations is extended to its action in \(K\GD \). This extension can be obtained, for example, in the following way. Note first, that for defining an action in \(K\GD \) it is sufficient to define an action in the ring \(K[\GD \cap \Z ^n]\). Define an action of \(K\R ^n\) on monomials \(x^\Go \in K[\Gd \cap \ZZ ^n]\) by formula \((\Ga _1,\dots ,\Ga _n)x^\Go =\Ga _1^{\Go _1}\dots \Ga _n^{\Go _n}\) and extend it to the whole ring \(K[\GD \cap \Z ^n]\) by linearity. Further, note that if \(V\subset \R ^n\) is a vector space, then the map \(\inj ^*:K\R ^n\to KV\) is a group homomorphism. Elements of kernel of \(\inj ^*:K\R ^n\to KC_\GD (\GD )\) act identically in \(K[\GD \cap \Z ^n]\). It allows to extract from the action of \(K\R ^n\) in \(K\GD \) an action of \(KC^\GD (\GD )\) in \(K\GD \), which extends the action of \(KC_\GD (\GD )\) in itself by translations.

With each face \(\GG \) of a cone \(\GD \) one associates (as with a smaller cone) a variety \(K\GG \) and a map \(\inj ^*:K\GD \to K\GG \). On the other hand there exists a map \(\inj _*:K\GG \to K\GD \) for which \(\inj ^*\circ \,\inj _*\) is the identity map \(K\GG \to K\GG \). Therefore, \(\inj _*\) is an embedding whose image is a retract of \(K\GD \). From the viewpoint of schemes the map \(\inj _*\) should be defined by the homomorphism \(K[\GD \cap \Z ^n]\to K[\GG \cap \Z ^n]\) which maps a Laurent polynomial \(\sum _{\Go \in \GD \cap \Z ^n}a_\Go x^\Go \) to its \(\GG \)-truncation \(\sum _{\Go \in \GG \cap \Z ^n}a_\Go x^\Go \). In terms of subvarieties of affine space, \(K\GG \) is the intersection of \(K\GD \) with the subspace \(y_{i_1}=y_{i_2}=\dots =y_{i_s}=0\), where \(y_{i_1},\dots ,y_{i_s}\) are the coordinates corresponding to generators of semigroup \(\GD \cap \Z ^n\) which do not lie in \(\GG \).

Varieties \(\inj _*(K\GG )\) with \(\GG \in \mathcal {G}_{\dim \GD -1}(\GD )\) cover \(K\GD \sminus \inj ^*(KC_\GD (\GD ))\). Images of algebraic tori \(KC_\GG (\GG )\) with \(\GG \in \mathcal {G}(\GD )\) under the composition

\[\begin {CD}KC_\GG (\GG )@> \inj ^* >> K\GG @> \inj ^* >> K\GD \end {CD}\]

of embeddings form a partition of \(K\GD \), which is a smooth stratification of \(K\GD \). Closure of the stratum \(\inj _*\inj ^*(KC_\GG (\GG ))\) in \(K\GD \) is \(\inj _*(K\GG )\). Below in the cases when it does not lead to confusion we shall identify \(K\GG \) with \(\inj _*K\GG \) and \(KC_\GG (\GG )\) with \(\inj _*\inj ^*KC_\GG (\GG )\) (i.e. we shall consider \(K\GG \) and \(KC_\GG (\GG )\) as lying in \(K\GD \)).

Figure 24 .

For any polyhedron \(\GD \) the toric variety \(K\GD \) is quasi-projective. If \(\GD \) is bounded, it is projective (see [MSDKK] and [Dan78] ).

A polyhedron \(\GD \subset \R ^n\) is said to be permissible if \(\dim \GD =n\), each face of \(\GD \) has a vertex and for any vertex \(\GG \in \mathcal {G}_0(\GD )\) the cone \(C_\GD (\GG )\) is simple. If polyhedron \(\GD \) is permissible then variety \(K\GD \) is nonsingular and it can be obtained by gluing affine spaces \(KC_\GD (\GG )\) with \(\GG \in \mathcal {G}_0(\GD )\). The gluing allows the following description. Let us associate with each cone \(C_\GD (\GG )\) where \(\GG \in \mathcal {G}_0(\GD )\) an automorphism \(f_\GG :K\R ^n\to K\R ^n\): if \(C_\GD (\GG )=\langle v_1,\dots ,v_n\rangle \) and \(v_i=(v_{i1},\dots ,v_{in})\) for \(i=1,\dots ,n\), then we put \(f_\GG (x_1,\dots ,x_n)=(x_1^{v_{11}}\dots x_n^{v_{1n}},\dots , x_1^{v_{n1}}\dots x_n^{v_{nn}})\). The variety \(K\GD \) is obtained by gluing to \(K\R ^n\) copies of \(K^n\) by maps \(\begin {CD}K\R ^n@> f_\GG >>K\R ^n\hookrightarrow K^n\end {CD}\) for all vertices \(\GG \) of \(\GD \). (Cf. Khovansky [Kho77] .)

The variety \(K\GD \) is defined by \(\GD \), but does not define it. Indeed, if \(\GD _1\) and \(\GD _2\) are polyhedra such that there exists a bijection \(\mathcal {G}(\GD _1)\to \mathcal {G}(\GD _2)\), preserving dimensions and inclusions and assigning to each face of \(\GD _1\) a parallel face of \(\GD _2\), then \(K\GD _1=K\GD _2\).

Denote by \(P^n\) the simplex of dimension \(n\) with vertices

\[(0,0,\dots ,0), (1,0,\dots ,0), (0,1,0,\dots ,0), \dots , (0,0,\dots ,1).\]

It is permissible polyhedron. \(KP^n\) is the n-dimensional projective space (this agrees with its usual notation).

Evidently, \(K(\GD _1\times \GD _2)=K\GD _1\times K\GD _2\). In particular, if \(\GD \subset \R ^2\) is a square with vertices \((0, 0)\), \((1, 0)\), \((0, 1)\) and \((1, 1)\), i.e. if \(\GD =P^1\times P^1\), then \(K\GD \) is a surface isomorphic to nonsingular projective surface of degree 2 (to hyperboloid in the case of \(K=\R ^2\)).

Polyhedra shown in Figure 25 define the following surfaces: \(K\GD _1\) is the affine plane with a point blown up; \(K\GD _2\) is projective plane with a point blown up (\(\R \GD _2\) is the Klein bottle); \(K\GD _3\) is the linear surface over \(KP^1\), defined by sheaf \(\mathcal {O}+\mathcal {O}(-2)\) (\(\R \GD _3\) is homeomorphic to torus).

Figure 25 .

The variety \(KC_\GD (\GD )\) is isomorphic to \(K\R ^{\dim \GD }\), open and dense in \(K\GD \), so \(K\GD \) can be considered as a completion of \(K\R ^{\dim \GD }\). Actions of \(KC_\GD (\GD )\) in affine parts \(KC_\GD (\GG )\) of \(K\GD \) correspond to each other and define an action in \(K\GD \) which is an extension of the action of \(KC_\GD (\GD )\) in itself by translations. Transformations of \(K\GD \) extending \(qh_{w,t}\) and \(S_w\) are denoted by the same symbols \(qh_{w,t}\) and \(S_w\).

The complement \(K\GD \sminus KC_\GD (\GD )\) is covered by \(K\GS \) with \(\GS \in \mathcal {G}(C_\GD (\GG ))\), \(\GG \in \mathcal {G}'(\GD )\) or, equivalently, by varieties \(KC_\GG (\GS )\) with \(\GS \in \mathcal {G}(\GG )\), \(\GG \in \mathcal {G}'(\GD )\). They comprise varieties \(K\GG \) with \(\GG \in \mathcal {G}'(\GD )\), which also cover \(K\GD \sminus KC_\GD (\GD )\). The varieties \(K\GG \) are situated with respect to each other in the same manner as the corresponding faces in the polyhedron: \(K(\GG _1\cap \GG _2)=K\GG _1\cap K\GG _2\). Algebraic tori \(KC_\GG (\GG )=K\GG \sminus \bigcup _{\GS \in \mathcal {G}'(\GG )}K\GS \) form partition of \(K\GD \), which is a smooth stratification; they are orbits of the action of \(KC_\GD (\GD )\) in \(K\GD \).

We shall say that a polyhedron \(\GD _2\) is richer than a polyhedron \(\GD _1\) if for any face \(\GG _2\in \mathcal {G}(\GD _2)\) there exists a face \(\GG _1\in \mathcal {G}(\GD _1)\) such that \(C_{\GD _2}(\GG _2)\supset C_{\GD _1}(\GG _1)\) (such a face \(\GG _1\) is automatically unique), and for each face \(\GG _1\in \mathcal {G}(\GD _1)\) the cone \(C_{\GD _1}(\GG _1)\) can be presented as the intersection of several cones \(C_{\GD _2}(\GG _2)\) with \(\GG _1\in \mathcal {G}(\GD _2)\). This definition allows a convenient reformulation in terms of dual cones: a polyhedron \(\GD _2\) is richer than polyhedron \(\GD _1\) iff the cones \(D^+C_{\GD _2}(\GG _2)\) with \(\GG _2\in \mathcal {G}(\GD _2)\) cover the set, which is covered by \(D^+C_{\GD _1}(\GG _1)\) with \(\GG _1\in \mathcal {G}(\GD _1)\), and the first covering is a refinement of the second.

Let a polyhedron \(\GD _2\) be richer than \(\GD _1\). Then the inclusions \(C_{\GD _1}(\GG _1)\hookrightarrow C_{\GD _2}(\GG _2)\) define for any \(\GG _2\in \mathcal {G}(\GD _2)\) a regular map \(\begin {CD}KC_{\GD _2}(\GG _2)@>\inj ^*>>KC_{\GD _1}(\GG _1)\hookrightarrow K\GD _1\end {CD}\). Obviously, these maps commute with the embeddings, by which \(K\GD _2\) and \(K\GD _1\) are glued from affine pieces, thus a regular map \(K\GD _2\to K\GD _1\) appears.

One can show (see, for example, [MSDKK] ) that for any polyhedron \(\GD _1\) there exists a richer polyhedron \(\GD _2\), defining a nonsingular toric variety \(K\GD _2\). Such a polyhedron is called a resolution of \(\GD _1\) (because it gives a resolution of singularities of \(K\GD _1\)). If \(\dim \GD =n\) (\(=\) the dimension of the ambient space \(\R ^n\)), then a resolution of \(\GD \) can be found among permissible polyhedra.

Proof.

Consider \(KC_\GD (\GG _2)\). It is a dense subset of \(K\GG _2\). Since \(C_{\GD (a)}(\GG _1)\subset C_\GD (\GG _2)\), there exists a monomial \(x^\Go \) such that \(\GD (x^\Go a)\) lies in \(C_\GD (\GG _2)\) and intersects its ridge exactly in the face obtained from \(\GG _1\). Since on \(K\GG _2\cap KC_\GD (\GG _2)\) all monomials, whose exponents do not lie on ridge \(C_{\GG _2}(\GG _2)\) of \(C_\GD (\GG _2)\), equal zero, it follows that the intersection \(\{x\in KC_\GD (\GG _2)\, |\, x^\Go a(x)=0\}\cap K\GG _2\) coincides with \(\{x\in KC_\GD (\GG _2)\, |\,[x^\Go a]^{C_{\GG _2}(\GG _2)}(x)=0\}\cap K\GG _2\). Note finally, that the latter coincides with \(V_{K\GG _2}(a_1)\).

The proof is analogous to the proof of the previous statement.

Denote by \(SV_{K\R ^n}(a)\) the set of singular points of \(V_{K\R ^n}(a)\), i.e. a set \(V_{K\R ^n}(a)\cap \bigcap _{i=1}^n V_{K\R ^n}(\frac {\partial a}{\partial x_i})\).

A Laurent polynomial \(a\) is said to be completely nondegenerate (over \(K\)) if, for any face \(\GG \) of its Newton polyhedron, \(SV_{K\R ^n}(a^\GG )\) is empty and, hence, \(V_{K\R ^n} (a^\GG )\) is a nonsingular hypersurface. A Laurent polynomial \(a\) is said to be peripherally nondegenerate if for any proper face \(\GG \) of its Newton polyhedron \(SV_{K\R ^n}(a^\GG )=\empt \).

It is not difficult to prove that completely nondegenerate L-polynomials form Zarisky open subset of the space of L-polynomials over \(K\) with a given Newton polyhedron, and the same holds true also for peripherally nondegenerate L-polynomials.

2.5.C.

If a Laurent polynomial \(a\) over \(K\) is completely nondegenerate and \(\GD \subset \R ^n\) is a resolution of its Newton polyhedron \(\GD (a)\) then the variety \(V_{K\GD }(a)\) is nonsingular and transversal to all \(K\GG \) with \(\GG \in \mathcal {G}'(\GD )\). See, for example, [Kho77] .

Theorem 2.5.C allows various generalizations related with possibilities to consider singular \(K\GD \) or only some faces of \(\GD (a)\) (instead of all of them). For example, one can show that if under the hypothesis of 2.5.A a truncation \(a^\GG \) of \(a\) is completely nondegenerate then under an appropriate understanding of transversality (in the sense of stratified space theory) \(V_{K\GD }(a)\) is transversal to \(K\GG _2\). Without going into discussion of transversality in this situation, I formulate a special case of this proposition, generalizing Theorem 2.5.C .

The proof of this proposition is a fragment of the proof of Theorem 2.5.C .

2.5.E.

(Corollary of 2.1.B and 2.1.C )) Let \(\GD \) and \(a\) be as in 2.5.A . Then for any vector \(w\in C_\GD (\GD )\) orthogonal to \(C_{\GD (a)}(\GD (a))\), a hypersurface \(V_{K\GD }(a)\) is invariant under transformations \(qh_{w,t}:K\GD \to K\GD \) and \(S_{(e^{\pi iw_1},\dots ,e^{\pi iw_n)}}:K\GD \to K\GD \) (the latter in the case of \(K=\R \) is defined only if \(w\in \Z ^n\)).

If \(\GD \) is a cone then \(\R _+\GD \) is defined as a subset of \(K\GD \) consisting of the points in which values of all monomials \(x^\Go \) with \(\Go \in \GD \cap \Z ^n\) are real and nonnegative. It is clear that for \(\GG \in \mathcal {G}'(\GD )\) the set \(\mathbb {R}_+ \GG \) coincides with \(\R _+\GD \cap K\GG \) and for cones \(\GD _1\subset \GD _2\) a preimage of \(\R _+\GD _1\) under \(\inj ^*:K\GD _2\to K\GD _1\) (see Section 2.3 ) is \(\R _+\GD _2\).

Now let \(\GD \) be an arbitrary polyhedron. Embeddings, by which \(K\GD \) is glued form \(KC_\GD (\GG )\) with \(\GG \in \mathcal {G}(\GD )\), embed the sets \(\R _+C_\GD (\GG )\) in one another; a space obtained by gluing from \(\R _+C_\GD (\GG )\) with \(\GG \in \mathcal {G}(\GD )\) is \(\R _+\GD \). It is clear that if \(\GG \in \mathcal {G}'\) then \(\R _+\GG =\R _+\GD \cap K\GG \).

\(\R _+\R ^n\) is the open positive orthant \(\{x\in \Bbb {RR}^n\,|\, x_1>0,\dots ,x_n>0\}\). It can be identified with the subgroup of quasi-homotheties of \(K\R ^n\): one assigns \(qh_{l(x)}\) to a point \(x\in \Bbb {R_+R}^n\).

If \(A^n=\{x\in \R ^n | x_1\ge 0,\dots ,x_n\ge 0\}\) then \(KA^n=K^n\) (cf. Section 2.3 ) and \(\R _+A^n=A^n\).

If \(P^n\) is the \(n\)-simplex with vertexes \((0,0,\dots ,0)\), \((1,0,\dots ,0)\), \((0,1,0, \dots , 0)\), …, \((0,0,\dots ,1)\), then \(KP^n\) is the \(n\)-simplex consisting of points of projective space with nonnegative real homogeneous coordinates.

The set \(\R _+\GD \) is invariant under quasi-homotheties. Orbits of action in \(\R _+\GD \) of the group of quasi-homotheties of \(\R _+\R ^n\) are sets \(\R _+C_\GG (\GG )\) with \(\GG \in \mathcal {G}(\GD )\). Orbit \(\R _+C_\GG (\GG )\) is homeomorphic to \(\R ^{\dim \GG }\) or, equivalently, to the interior of \(\GG \). Closures \(\R _+\GG \) of \(\R _+C_\GG (\GG )\) intersect one another in the same manner as the corresponding faces: \(\R _+\GG _1\cap \R _+\GG _2=\R _+(\GG _1\cap \GG _2)\). From this and from the fact that \(\R _+\GG \) is locally conic (see [Loj64] ) it follows that \(\R _+\GD \) is homeomorphic, as a stratified space, to \(\GD \). However, there is an explicitly constructed homeomorphism. It is provided by the Atiyah moment map [Ati81] and in the case of bounded \(\GD \) can be described in the following way.

Choose a collection of points \(\Go _1,\dots ,\Go _k\) with integer coordinates, whose convex hull is \(\GD \). Then for \(\GG \in \mathcal {G}(\GD )\) and \(\Go _0\in \GG \smallsetminus \partial \GG \) cone \(C_\GD (\GG )\) is \(\langle \Go _1-\Go _0,\dots ,\Go _k-\Go _0 \rangle \). For \(y\in KC_\GD (\GG )\) denote by \(y^\Go \) a value of monomial \(x^\Go \) where \(\Go \in C_\GD (\GG )\cap \Z ^n\) at this point. Put

\[M(y)=\dfrac {\sum _{i=1}^{k}|y^{\Go _i-\Go _0}|\Go _i} {\sum _{i=1}^{k}{|y^{\Go _i-\Go _0}|}}\in \R ^n.\]

Obviously \(M(y)\) lies in \(\GD \), does non depend on the choice of \(\Go _0\) and for \(y\in KC_\GD (\GG _1)\cap KC_\GD (\GG _2)\) does not depend on what face, \(\GG _1\) or \(\GG _2\), is used for the definition of \(M(y)\). Thus a map \(M:K\GD \to \GD \) is well defined. It is not difficult to show that \(M|_{\R _+\GD }:\R _+\GD \to \GD \) is a stratified homeomorphism.

\(\R _+\GD \) is a fundamental domain for the natural action of \(U_K^n\) in \(K\GD \), i.e. its intersection with each orbit of the action consists of one point.

For a point \(x\in \R _+C_\GG \GG \) where \(\GG \in \mathcal {G}(\GD )\), the stationary subgroup of action of \(U_K^n\) consists of transformations \(S_{(e^{\pi iw_1},\dots ,e^{\pi iw_n})}\), where vector \((w_1,\dots ,w_n)\) is orthogonal to \(C_\GG (\GG )\). In particular, if \(\dim \GG =n\) then the stationary subgroup is trivial. If \(\dim \GG =n-r\) then it is isomorphic to \(U_K^r\). Denote by \(U_\GG \) a subgroup of \(U_K^n\) consisting of elements \((e^{\pi iw_1},\dots ,e^{\pi iw_n})\) with \((w_1,\dots ,w_n)\perp C_\GG (\GG )\).

Define a map \(\rho :\R _+\GD \times U_K^n\to K\GD \) by formula \((x,y)\mapsto S_y(x)\). It is surjection and we know the partition of \(\R _+\GD \times U_K^n\) into preimages of points. Since \(\rho \) is proper and \(K\GD \) is locally compact and Hausdorff, it follows that \(K\GD \) is homeomorphic to the quotientspace of \(\R _+\GD \times U_K^n\) with respect to the partition into sets \(x\times yU_\GG \) with \(x\in \R _+C_\GG (\GG )\), \(y\in U_K^n\).

Consider as an example the case of \(K=\R \) and \(n=2\). Let a polyhedron \(\GD \) lies in the open positive quadrant. We place \(\GD \times U_\R ^2\) in \(\R ^2\) identifying \((x,y)\in \GD \times U_\R ^2\) with \(S_y(x)\in \R ^2\). \(\Bbb R_+\GD \times U_\R ^2\) is homeomorphic \(\GD \times U_\R ^2\), so the surface \(\R \GD \) can be obtained by an appropriate gluing (namely, by transformations taken from \(U_\GG \)) sides of four polygons consisting \(\GD \times U_\R ^2\). Figure 26 shows what gluings ought to be done in three special cases.

Figure 26 .

\[h(x,y,z)=(pr_{\R _+\GG }h(x,y),\,zpr_{U_K^n}h(x,y)).\]

For \(h\) one can take, for example, \((M|_{\R _+\GD })^-\).

A pair consisting of \(\GD \times U_K^n\) and its subset \(\upsilon \) which is the preimage of \(V_{K\GD }(a)\) under

\[\begin {CD}\GD \times U_K^n@>h\times \id >>\R _+\GD \times U_K^n@>\rho >>K\GD \end {CD}\]

is called a (nonreduced) \(K\)-chart of L-polynomial \(a\).

It is clear that the set \(\upsilon \) is invariant under transformations \(\id \times S\) with \(S\in U_\GD \) and its intersection with \(\GG \times U_K^n\), where \(\GG \in \mathcal {G}'(\GD )\) is invariant under transformations \(\id \times S\) with \(S\in U_\GG \).

As it follows from 2.5.A , if \(\GG \) is a face of \(\GD \), and \((\GD \times U_K^n, \,\upsilon )\) is a nonreduced \(K\)-chart of L-polynomial \(a\), then \((\GG \times U_K^n, \,\upsilon \cap (\GG \times U_K^n))\) is a nonreduced \(K\)-chart of L-polynomial \(a^\GG \).

A nonreduced \(K\)-chart of Laurent polynomial \(a\) is unique up to homeomorphism \(\GD \times U_K^n\to \GD \times U_K^n\), satisfying the following two conditions:

1. it map \(\GG \times y\) with \(\GG \in \mathcal {G}(\GD )\) and \(y\in U_K^n\) to itself and
2. its restriction to \(\GG \times U_K^n\) with \(g\in \mathcal {G}(\GD )\) commutes with transformations \(\id \times S:\GG \times U_K^n\to \GG \times U_K^n\) where \(S\in U_\GG \).

In the case when \(a\) is a usual polynomial, it is convenient to place its \(K\)-chart into \(K^n\). For this, consider a map \(A^n\times U_K^n\to K^n:(x,y)\mapsto S_y(x)\). Denote by \(\GD _K(a)\) the image of \(\GD (a)\times U_K^n\) under this map. Call by a (reduced) \(K\)-chart of \(a\) the image of a nonreduced \(K\)-chart of \(a\) under this map. The charts of peripherally nondegenerate real polynomial in two variables introduced in Section 1.3 are \(\R \)-charts in the sense of this definition.

Proof.

1. all the restrictions \(\nu |_{\GD _i}\) are linear;
2. if the restriction of \(\nu \) to an open set is linear then this set is contained in one of \(\GD _i\);
3. \(\nu (\GD \cap \Z ^n)\subset \Z \).

Figure 27 .

Let \(a_1,\dots ,a_s\) be Laurent polynomials over \(K\) in \(n\) variables with \(\GD (a_i)=\GD \). Let \(a_i^{\GD _i\cap \GD _j}=a_j^{\GD _i\cap \GD _j}\) for any \(i\), \(j\). Then, obviously, there exists an unique L-polynomial \(a\) with \(\GD (a)=\GD \) and \(a^{\GD _i}=a_i\) for \(i=1,\dots ,s\). If \(a(x_1,\dots ,x_n)=\sum _{\Go \in \Z ^n}a_\Go x^\Go \), we put \(b(x,t)=\sum _{\Go \in \Z ^n}a_\Go x^\Go t^{\nu (\Go )}\). This L-polynomial in \(n+1\) variables is considered below also as a one-parameter family of L-polynomials in \(n\) variables. Therefore let me introduce the corresponding notation: put \(b_t(x_1,\dots ,x_n)=b(x_1,\dots ,x_n,t)\). L-polynomials \(b_t\) are said to be obtained by patchworking L-polynomials \(a_1,\dots ,a_s\) by \(\nu \) or, briefly, \(b_t\) is a patchwork of L-polynomials \(a_1,\dots ,a_s\) by \(\nu \).

Proof.

By singularity of a hypersurface \(V_G(\varphi )\) at the point \(x_0\in V_G(\varphi )\) we mean the class of germs of hypersurfaces which are diffeomorphic to the germ of \(V_G(\varphi )\) at \(x_0\). In other words, hypersurfaces \(V_G(\varphi )\) and \(V_H(\psi )\) have the same singularity at points \(x_0\) and \(y_0\), if there exist neighborhoods \(M\) and \(N\) of \(x_0\) and \(y_0\) such that the pairs \((M, V_M(\varphi ))\), \((N, V_N(\psi ))\) are diffeomorphic. When considering a singularity of hypersurface at a point \(x_0\), to simplify the formulas we shall assume that \(x_0=0\).

The multiplicity or the Milnor number of a hypersurface \(V_G(\varphi )\) at \(0\) is the dimension

\[dim_KK[[x_1,\dots ,x_n]]/(\p f/\p x_1,\dots , \p f/\p x_n)\]

of the quotient of the formal power series ring by the ideal generated by partial derivatives \(\p f/\p x_1,\dots ,\p f/\p x_n\) of the Taylor series expansion \(f\) of the function \(\varphi \) at \(0\). This number is an invariant of the singularity (see [AVGZ82] ). If it is finite, then we say that the singularity is of finite multiplicity.

If the singularity of \(V_G(\Gf )\) at \(x_0\) is of finite multiplicity, then this singularity is isolated, i.e. there exists a neighborhood \(U\subset K^n\) of \(x_0\), which does not contain singular points of \(V_G(\varphi )\). If \(K=\CC \) then the converse is true: each isolated singularity of a hypersurface is of finite multiplicity. In the case of isolated singularity, the boundary of a ball \(B\subset K^n\), centered at \(x_0\) and small enough, intersects \(V_G(\varphi )\) only at nonsingular points and only transversely, and the pair \((B, V_B(\varphi ))\) is homeomorphic to the cone over its boundary \((\p B, V_{\p B}(\varphi ))\) (see [Mil68] , Theorem 2.10). In such a case the pair \((\p B, V_{\p B}(\varphi ))\) is called the link of singularity of \(V_G(\varphi )\) at \(x_0\).

The following Theorem shows that the class of singularities of finite multiplicity of analytic hypersurfaces coincides with the class of singularities of finite multiplicity of algebraic hypersurfaces.

If the family \(\Gf _t\) with \(t\in [0,t_0]\) evolves the singularity of the hypersurface \(V_G(\Gf _0)\) at \(x_0\), then there exists a ball \(B\subset K^n\) centered at \(x_0\) such that

1. for \(t\in [0,t_0]\) the sphere \(\p B\) intersects \(V_G(\Gf _t)\) only in nonsingular points of the hypersurface and only transversely,
2. for \(t\in (0,t_0]\) the ball \(B\) contains no singular point of the hypersurface \(V_G(\Gf _t)\),
3. the pair \((B, V_B(\Gf _0))\) is homeomorphic to the cone over its boundary \((\p B\), \(V_{\p B}(\Gf _0))\).

Then the family of pairs \((B, V_B(\Gf _t))\) with \(t\in [0,t_0]\) is called an evolving of the germ of \(V_G(\Gf _0)\) in \(x_0\). (Following the standard terminology of the singularity theory, it would be more correct to say not a on family of pairs, but rather a family of germs or even germs of a family; however, from the topological viewpoint, which is more natural in the context of the topology of real algebraic varieties, the distinction between a family of pairs satisfying 5.2 and 5.2 and the corresponding family of germs is of no importance, and so we shall ignore it.)

Conditions 5.2 and 5.2 imply existence of a smooth isotopy \(h_t:B\to B\) with \(t\in (0,t_0]\), such that \(h_{t_0}=\id \) and \(h_t(V_B(\Gf _{t_0}))=V_B(\Gf _t)\), so that the pairs \((B,V_B(\Gf _t))\) with \(t\in (0,t_0]\) are homeomorphic to each other.

Given germs determining the same singularity, a evolving of one of them obviously corresponds to a diffeomorphic evolving of the other germ. Thus, one may speak not only of evolvings of germs, but also of evolvings of singularities of a hypersurface.

The following three topological classification questions on evolvings arise.

Smoothings \((B,V_B(\Gf _t))\) with \(t\in [0,t_0]\) and \((B',V_{B'}(\Gf '_t))\) with \(t\in [0,t'_0]\) are said to be topologically equivalent if there exists an isotopy \(h_t:B\to B'\) with \(t\in [0, \min (t_0,t'_0)]\), such that \(h_0\) is a diffeomorphism and \(V_{B'}(\Gf '_t)=h_tV_B(\Gf _t)\) for \(t\in [0, \min (t_0,t'_0)]\).

Obviously, 5.2.B is a refinement of 5.2.A . In turn, 5.2.C is more refined than 5.2.B , since in 5.2.C we are interested not only in the type of the pair obtained in result of the evolving, but also the manner in which the pair is attached to the link of the singularity.

In the case \(K=\R \) these questions have been answered in literature only for several simplest singularities.

In the case \(K=\C \) a evolving of a given singularity is unique from each of the three points of view, and there is an extensive literature (see, for example, [GZ77] ) devoted to its topology (i.e., questions 5.2.A and 5.2.B ).

By the way, if we want to get questions for \(K=\C \) which are truly analogous to questions 5.2.A — 5.2.C for \(K=\R \), then we have to replace evolvings by deformations with nonsingular fibers and one-dimensional complex bases, and the variety \(V_B(\Gf _t)\) and the pairs \((B, V_B(\Gf _t))\) have to be considered along with the monodromy transformations. It is reasonable to suppose that there are interesting connections between questions 5.2.A — 5.2.C for a real singularity and their counter-parts for the complexification of the singularity.

Denote by \(\Pi _K(\Gf )\) a part of \(K^n\) bounded by \(\Sigma _K(\Gf )\). It can be presented as a cone over \(\Sigma _K(\Gf )\) with vertex at zero.

One can choose the isotopy \(h_t:B\to B\), \(t\in (0,t_0]\) such that its restriction to \(\p B\) can be extended to an isotopy \(h'_t:\p B\to \p B\) with \(t\in [0,t_0]\) (i.e., extended for \(t=0\)).

We shall call the pair \((\Pi _K(\Gf )\), \(\tau )\) a chart of evolving \((B\), \(V_B(\Gf _t))\), \(t\in [0,t_0]\), if there exists a homeomorphism \((\Pi _K(\Gf )\), \(\tau )\to (B\), \(V_B(t_0))\), whose restriction \(\Sigma _K(\Gf )\to \p B\) is the composition \(\begin {CD}\Sigma _K(\Gf )@>g>>\p B@>h'_0>>\p B\end {CD}\). One can see that the boundary \((\p \Pi _K(\Gf )\), \(\p \tau )\) of a chart of evolving is a chart \((\Sigma _K(\Gf )\), \(\gamma )\) of the germ of the hypersurface at zero, and a chart of evolving is a pair obtained by evolving which is glued to \((\Sigma _K(\Gf )\), \(\gamma )\) in natural way. Thus that the chart of an evolving describes the evolving up to topological equivalence.

Let \(a_1, \dots , a_s\) be completely nondegenerate polynomials over \(K\) in \(n\) variables with \(\Int \GD (a_i)\cap \Int \GD (a_j)=\varnothing \) and \(a_i^{\GD (a_i)\cap \GD (a_j)}=a_j^{\GD (a_i)\cap \GD (a_j)}\) for \(i\ne j\). Let \(\bigcup _{i=1}^s\GD (a_i)\) be the polyhedron bounded by the coordinate axes and Newton diagram \(\GG (f)\). Let \(a_i^{\GD (a_i)\cap \GD (f)}=f^{\GD (a_i)\cap \GD (f)}\) for \(i=1,\dots , s\). Let \(\nu :\bigcup _{i=1}^s\GD (a_i)\to \R \) be a nonnegative convex function which is equal to zero on \(\GG (f)\) and satisfies conditions 4.1 , 4.1 , 4.1 of Section 4.1 with polyhedra \(\GD (a_1)\), …, \(\GD (a_s)\). Then polynomials \(a_1, \dots , a_s\) can be "glued to \(\Gf \) by \(\nu \)" in the following way generalizing patchworking L-polynomials of Section 4.1 .

Denote by \(a\) the polynomial defined by conditions \(a^{\GD (a_i)}=a_i\) for \(i=1, \dots , s\) and \(a^{\bigcup _{i=1}^{s}\GD (a_i)}=a\). If \(a(x)=\sum _{\Go \in \Z ^n}a_\Go x^\Go \) then we put

\[\Gf _t(x)=\Gf (x)+(\sum _{\Go \in \Z ^n}a_\Go x^\Go t^{\nu (\Go )})-a^{\GG (f)}x.\]

In the case when \(\Gf \) is a polynomial, Theorem 5.4.A is a slight modification of a special case of Theorem 4.3.A . Proof of 4.3.A is easy to transform to the proof of this version of 5.4.A . The general case can be reduced to it by Tougeron Theorem, or one can prove it directly, following to scheme of proof of Theorem 4.3.A .

We shall call the evolvings obtained by the scheme described in this Section patchwork evolvings.

We need tubular neighborhoods mainly for formalizing a notion of approximation of a submanifold by a submanifold. A manifold presented as the image of a smooth section of the tubular fibration of a tubular \(\Ge \)-neighborhood of \(M\) can be considered as sufficiently close to \(M\): it is naturally isotopic to \(M\) by an isotopy moving each point at most by \(\Ge \).

We shall consider the space \(\R ^n\times U_K^n\) as a flat Riemannian manifold with metric defined by the standard Euclidian metric of \(\R ^n\) in the case of \(K=\R \) and by the standard Euclidian metric of \(\R ^n\) and the standard flat metric of the torus \(U_{\C }^n=(S^1)^n\) in the case of \(K=\C \).

An \(\Ge \)-sufficiency of truncations of Laurent polynomial defined below and the whole theory related with this notion presuppose that it has been chosen a class of tubular neighborhoods of smooth submanifolds of \(\R ^n\times U_K^n\) invariant under translations \(T_\Go \times \id _{U_K^n}\) and that for any two tubular neighborhoods \(N\) and \(N'\) of the same \(M\), which belong to this class, restrictions of tubular fibrations \(p:N\to M\) and \(p':N'\to M\) to \(N\cap N'\) coincide. One of such classes is the collection of all normal tubular neighborhoods, i.e. tubular neighborhoods with fibers consisting of segments of geodesics which start from the same point of the submanifold in directions orthogonal to the submanifold. Another class, to which we shall turn in Sections 6.7 and 6.8 , is the class of tubular neighborhoods whose fibers lie in fibers \(\R ^{n-1}\times t\times U_K^{n-1}\times s\) of \(\R ^n\times U_K^n\) and consist of segments of geodesics which are orthogonal to intersections of the corresponding manifolds with these \(\R ^{n-1}\times t\times U_K^{n-1}\times s\). The intersection of such a tubular neighborhood of \(M\) with the fiber \(\R ^{n-1}\times t\times U_K^{n-1}\times s\) is a normal tubular neighborhood of \(M\cap (\R ^{n-1}\times t\times U_K^{n-1}\times s)\) in \(\Bbb R^{n-1}\times t\times U_K^{n-1}\times s\). Of course, only manifolds transversal to \(\R ^{n-1}\times t\times U_K^{n-1}\times s\) have tubular neighborhoods of this type.

Introduce a norm in vector space of Laurent polynomials over \(K\) on \(n\) variables:

\[||\sum _{\Go \in \Z ^n}a_\Go x^\Go ||=\max \{|a_\Go |\,|\,\Go \in \Z ^n\}.\]

Let \(\GG \) be a subset of \(\R ^n\) and \(\Ge \) a positive number. Let \(a\) be a Laurent polynomial over \(K\) in \(n\) variables and \(U\) a subset of \(K\R ^n\). We shall say that in \(U\) the truncation \(a^\GG \) is \(\Ge \)-sufficient for \(a\) (with respect to the chosen class of tubular neighborhoods), if for any Laurent polynomial \(b\) over \(K\) satisfying the conditions \(\GD (b)\subset \GD (a)\), \(b^\GG =a^\GG \) and \(||b-b^\GG ||\le ||a-a^\GG ||\) (in particular, for \(b=a\) and \(b=a^\GG \)) the following condition takes place:

1. \(U\cap SV_{K\R ^n}(b)=\varnothing \),
2. the set \(la(U\cap V_{K\R ^n}(b))\) lies in a tubular \(\Ge \)-neighborhood \(N\) (from the chosen class) of \(la(V_{K\R ^n}(a^\GG )\sminus SV_{K\R ^n}(a^\GG ))\) and
3. \(la(U\cap V_{K\R ^n}(b))\) can be extended to the image of a smooth section of the tubular fibration \(N\to la(V_{K\R ^n}(a^\GG )\smallsetminus SV_{K\R ^n}(a^\GG ))\).

The \(\Ge \)-sufficiency of \(\GG \)-truncation of Laurent polynomial \(a\) in \(U\) means, roughly speaking, that monomials which are not in \(a^\GG \) have a small influence on \(V_{K\R ^n}(a)\cap U\).

6.1.A.

If \(a^\GG \) is \(\Ge \)-sufficient for \(a\) in open sets \(U_i\) with \(i\in \mathcal J\), then it is \(\Ge \)-sufficient for \(a\) in \(\bigcup _{i\in \mathcal {J}}U_i\) too.

Standard arguments based on Implicit Function Theorem give the following Theorem.

6.1.B.

If a set \(U\subset K\R ^n\) is compact and contains no singular points of a hypersurface \(V_{K\R ^n}(a)\), then for any tubular neighborhood \(N\) of \(V_{K\R ^n}(a)\sminus SV_{K\R ^n}(a)\) and any polyhedron \(\GD \supset \GD (a)\) there exists \(\Gd >0\) such that for any Laurent polynomial \(b\) with \(\GD (b)\subset \GD \) and \(||b-a||<\Gd \) the hypersurface \(V_{K\R ^n}(b)\) has no singularities in \(U\), intersection \(U\cap V_{K\R ^n}(b)\) is contained in \(N\) and can be extended to the image of a smooth section of a tubular fibration \(N\to V_{K\R ^n}(a)\sminus SV_{K\R ^n}(a)\).

From this the following proposition follows easily.

6.1.C.

If \(U\in K\R ^n\) is compact and \(a^\GG \) is \(\Ge \)-sufficient truncation of \(a\) in \(U\), then for any polyhedron \(\GD \supset \GD (a)\) there exists \(\Gd >0\) such that for any Laurent polynomial \(b\) with \(\GD (b)\subset \GD \), \(b^\GG =a^\GG \) and \(||b-a||<\Gd \) the truncation \(b^\GG \) is \(\Ge \)-sufficient in \(U\).

In the case of \(\GG =\GD (a)\) proposition 6.1.C turns to the following proposition.

6.1.D.

If a set \(U\subset K\R ^n\) is compact and contains no singular points of \(V_{K\R ^n}(a)\) and \(la (V_{K\R ^n}(a))\) has a tubular neighborhood of the chosen type, then for any \(\Ge >0\) and any polyhedron \(\GD \supset \GD (a)\) there exists \(\Gd >0\) such that for any Laurent polynomial \(b\) with \(\GD (b)\subset \GD \), \(||b-a||<\Gd \) and \(b^{\GD (a)}=a\) the truncation \(b^{\GD (a)}\) is \(\Ge \)-sufficient in \(U\).

The following proposition describes behavior of the \(\Ge \)-sufficiency under quasi-homotheties.

The proof follows from comparison of the definition of \(\Ge \)-sufficiency and the following two facts. First, it is obvious that

\[qh_{w,t}(U)\cap V_{K\R ^n}(b)=qh_{w,t}(U\cap qh^{-1}_{w,t}(V_{K\R ^n}(b)))=qh_{w,t}(U\cap V_{K\R ^n}(b\circ qh_{w,t})),\]

and second, the transformation \(T_{(\ln t)w}\times \id _{U_K^n}\) of \(\R ^n\times U_K^n\) corresponding, by 2.1.A , to \(qh_{w,t}\) preserves the chosen class of tubular \(\Ge \)neighborhoods .

For \(A\subset \R ^n\) and \(\rho >0\) denote by \(\frak N_\rho (A)\) the set \(\{x\in \R ^n|dist(x,A)<\rho \}\).

For \(A,B\subset \R ^n\) and \(\lambda \in \R \) the sets \(\{x+y\,|\,x\in A,y\in B\}\) and \(\{\lambda x\,|\,x\in A\}\) are denoted, as usually, by \(A+B\) and \(\lambda A\).

Let \(a\) be a Laurent polynomial in \(n\) variables, \(\Ge \) a positive number and \(\GG \) a face of the Newton polyhedron \(\GD =\GD (a)\).

Proof.

Proof.

This proposition is proved similarly to 6.2.B , but with the following difference: the reference to Theorem 6.1.D is replaced by a reference to Theorem 6.1.C .

Proof.

By 6.2.C , for any face \(\GS \) of \(\GD \) with \(\dim \GS =\dim \GD -1\) and \(\GG \subset \p \GS \) there exists a vector \(\Go _\GS \in DC_\GD ^-(\GS )\) such that the truncation \(a^\GG \) is \(\Ge \)-sufficient for \(a\) in \(qh_{\Go _\GS }(C)\), and, hence, by 6.2.A , in \(qh_{\Go _\GS +\Cl DC^-_\GD (\GG )}(\Int C)\). Choose such \(\Go _\GS \) for each \(\GS \) with \(\dim \GS =\dim \GD -1\) and \(\GG \subset \p \GS \). Obviously, the sets \(\Go _\GS +\Cl DC^-_\GD (\GG )\) cover the whole closure of the cone \(DC_\GD ^-(\GG )\) besides some neighborhood of its top, i.e. the cone \(DC^-_\GD (\GD )\); in other words, there exists a number \(\Gd \) such that \(\bigcup _\GS (\Go _\GS +\Cl DC_\GD ^-(\GG ))\supset \Cl DC_\GD ^-(\GG )\sminus \frak N_\Gd DC_\GD ^-(\GD )\). Hence, \(a^\GG \) is \(\Ge \)-sufficient for \(a\) in \(qh_{\Cl DC^-_\GD (\GG )\sminus \frak N_\Gd DC^-_\GD (\GD )}(\Int C)\subset \bigcup _\GS qh_{\Go _\GS +\Cl DC_\GD ^-(\GG )}(\Int C)\).

\[\frak N_{\Gf (\GG )}(DC_\GD ^-(\GG ))\sminus \bigcup _{\GS \in \mathcal {G}(\GD ), \‚\GG \in \mathcal {G}(\GS )} \frak N_{\Gf (\GS )}(DC_\GD ^-(\GS )).\]

It is clear that the sets \(D_{\GD ,\Gf }(\GG )\) with \(\GG \in \mathcal {G}(\GD )\) cover \(\R ^n\). Among these sets only sets corresponding to faces of the same dimension can intersect each other. In some cases (for example, if \(\Gf ^\GG \) grows fast enough when \(\dim \GG \) grows) they do not intersect and then \(\{D_{\GD ,\Gf }(\GG )\}_{\GG \in \mathcal {G}(\GD )}\) is a partition of \(\R ^n\).

Let \(a\) be a Laurent polynomial over \(K\) in \(n\) variables and \(\Ge \) be a positive number. A function \(\Gf :\mathcal {G}(\GD (a))\to \R ^n\) is said to be describing domains of \(\Ge \)-sufficiency for \(a\) (with respect to the chosen class of tubular neighborhoods) if for any proper face \(\GG \in \mathcal {G}(\GD (a))\), for which truncation \(a^\GG \) is completely non-degenerate and the hypersurface \(la(V_{K\R ^n}(a^\GG ))\) has a tubular neighborhood of the chosen class, the truncation \(a^\GG \) is \(\Ge \)-sufficient for \(a\) in some neighborhood of \(l^{-1}(DC_{\GD (a),\Gf }(\GG ))\).

In particular, if \(a\) is peripherally nondegenerate Laurent polynomial over \(K\) in \(n\) variables and \(\dim \GD (a)=n\) then for any \(\Ge >0\) there exists a compact set \(C\subset K\R ^n\) such that \(K\R ^n\sminus C\) is covered by regions in which truncations of \(a^{\p \GD (a)}\) are \(\Ge \)-sufficient for \(a\) with respect to class of normal tubular neighborhoods. In other words, under these conditions behavior of \(V_{K\R ^n}(a)\) outside \(C\) is defined by monomials of \(a^{\p \GD (a)}\).

If \(\dim \GD (a)=0\) then \(a\) is monomial and \(V_{K\R ^n}(a)=\varnothing \). Thus for any \(\Ge >0\) any function \(\Gf :\mathcal {G}(\GD (a))\to \R \) describes domains of \(\Ge \)-sufficiency for \(a\).

Induction step follows obviously from the following Theorem.

Proof.

The pair \((\GD \times U_K^n,\) \(\upsilon )\) consisting of the product \(\GD \times U_K^n\) and its subset \(\upsilon \) is a \(K\)-chart of a Laurent polynomial \(a\) if:

1. there exists a homeomorphism \(h:(\Cl D_{\GD ,\Gf }(\GD )\cap V)\times U_K^n\to \GD \times U_K^n\) such that \(h((Cl\phantom {a}D_{\GD ,\Gf }(\GD )\cap V)\times y)=\GD \times y\) for \(y\in U_K^n\),

   \[\upsilon =h(la V_{K\Bbb R^n}(a)\cap (\Cl D_{\GD ,\Gf }(\GD )\cap V)\times U_K^n\]

   and for each face \(\GG \) of \(\GD \) the set \(h((\Cl D_{\GD ,\Gf }(\GD ) \cap D_{\GD ,\Gf }(\GG )\cap V)\times U_K^n)\) lies in the product of the star \(\bigcup _{\GG \in \mathcal {G}(\GS )\\\GS \in \mathcal {G}(\GD )\sminus \{\GD \}}\GS \) of \(\GG \) to \(U_K^n\);

2. for any vector \(\Go \in \R ^n\), which is orthogonal to \(V\) and, in the case of \(K=\R \), is integer, the set \(\upsilon \) is invariant under transformation \(\GD \times U_K^n\to \GD \times U_K^n\) defined by formula \((x,\) \((y_1,\)…, \(y_n))\mapsto (x,\) \((e^{\pi i\Go _1}y_1,\) …, \(e^{\pi i\Go _n}y_n))\);

3. for each face \(\GG \) of \(\GD \) the pair \((\GG \times U_K^n,\) \(\upsilon \cap (\GG \times U_K^n))\) is a \(K\)-chart of Laurent polynomial \(a^\GG \).

The definition of the chart of a Laurent polynomial, which, as I believe, is clearer than the description given here, but based on the notion of toric completion of \(K\R ^n\), is given above in Section 3.3 . I restrict myself to the following commentary of conditions 6.6 – 6.6 .

The set \((\Cl D_{\GD ,\Gf }(\GD )\cap V)\times U_K^n\) contains, by 6.3.A , a deformation retract of \(la V_{K\R ^n}(a)\). Thus, condition 6.6 means that \(\upsilon \) is homeomorphic to a deformation retract of \(V_{K\R ^n}(a)\). The position of \(\upsilon \) in \(\GD \times U_K^n\) contains, by 6.6 and 6.6 , a complete topological information about behavior of this hypersurface outside some compact set. The meaning of 6.6 is in that \(\upsilon \) has the same symmetries as, according to 2.1.C , \(V_{K\R ^n}(a)\) has.

\[V_{K\R ^n}(b_t)=i^{-1}_t V_{K\R ^{n+1}}(b).\]

This allows to take advantage of results of the previous Section for study of \(V_{K\R ^n}(b_t)\) as \(t\to 0\). For sufficiently small \(t\) the image of embedding \(i_t\) is covered by regions of \(\Ge \)-sufficiency of truncation \(b^{\tilde \GG }\), where \(\tilde \GG \) runs over the set of faces of graph of \(\nu \), and therefore the hypersurface \(V_{K\R ^n}(b_t)\) turns to be composed of pieces obtained from \(V_{K\R ^n}(a_i)\) by appropriate quasi-homotheties.

I preface the formulation describing in detail the behavior of \(V_{K\R ^n}(b_t)\) with several notations.

Denote the Newton polyhedron \(\GD (b)\) of Laurent polynomial \(b\) by \(\tilde \GD \). It is clear that \(\tilde \GD \) is the convex hull of the graph of \(\nu \). Denote by \(\mathcal {G}\) the union \(\bigcup _{i=1}^s\mathcal {G}(\GD _i)\). For \(\GG \in \mathcal {G}\) denote by \(\tilde \GG \) the graph \(\nu |_\GG \). It is clear that \(\tilde \GG \in \mathcal {G}(\tilde \GD )\) and hence an injection \(\GG \to \tilde \GG :\mathcal {G}\to \mathcal {G}(\tilde \GD )\) is defined.

For \(t>0\) denote by \(j_t\) the embedding \(\R ^n\to \R ^{n+1}\) defined by the formula \(j_t(x_1, \dots ,x_n)=(x_1, \dots ,x_n,\,\ln t)\). Let \(\psi :\mathcal {G}\to \R \) be a positive function, \(t\) be a number from interval \((0,1)\). For \(\GG \in \mathcal G\) denote by \(\mathcal {E}_{t,\psi }(\GG )\) the following subset of \(\R ^n\):

\[\frak N_{\psi (\GG )}j_t^{-1}(DC^-_{\tilde \GD }(\tilde \GG ))\sminus \bigcup _{\GS \in \mathcal {G}, \ \GG \in \mathcal {G}(\GS )}\frak N_{\Gf (\GS )}j_t^{-1}(DC^-_{\tilde \GD }(\tilde \GS )).\]

Denote the gradient of restriction of \(\nu \) on \(\GG \in \mathcal G\) by \(\nabla (\GG )\). The truncation \(b_t^\GG \), obviously, equals \(a^\GG \circ qh_{\nabla (\GG ),t}\). In particular, \(b_t^{\GD _i}=a_i\circ qh_{\nabla (\GD _i),t}\) and, hence,

\[V_{K\R ^n}(b_t^{\GD _i})=qh_{\nabla (\GD _i),t^{-1}}(V_{K\Bbb R^n}(a_i)).\]

In the domain, where \(b^\GG _t\) is \(\Ge \)-sufficient for \(b_t\), the hypersurfaces \(la V_{K\R ^n}(b_t)\) and \(la V_{K\R ^n}(b_t^{\GD _i})\) with \(\GD _i\supset \GG \) lie in the same normal tubular \(\Ge \)-neighborhood of \(la V_{K\R ^n}(b_t^\GG )\) and, hence, are isotopic by an isotopy moving points at most on \(2\Ge \). Thus, according to 6.7.A , for \(t\le t_0\) to the space \(K\R ^n\) is covered by regions in which \(V_{K\R ^n}(b_t)\) is approximated by \(qh_{\nabla (\GD _i),t^{-1}}(V_{K\R ^n}(a_i))\).

\[j_t^{-1}(\frak N_{c\Gf (\tilde \GG )}(DC^-_{\tilde \GD }(\tilde \GG ))\sminus \bigcup _{\tilde \GS \in \mathcal {G}(\tilde \GD ), \ \tilde \GG \in \mathcal {G}(\tilde \GS )}\frak N_{\Gf (\tilde \GS )}(DC^-_{\tilde \GD }(\tilde \GS ))),\]

and thus from \(\Ge \)-sufficiency of \(b^{\tilde \GG }\) for \(b\) in some neighborhood of

\[l^{-1}(\frak N_{c\Gf (\tilde \GG )}(DC^-_{\tilde \GD }(\tilde \GG ))\sminus \bigcup _{\tilde \GS \in \mathcal {G}(\tilde \GD ), \ \tilde \GG \in \mathcal {G}(\tilde \GS )}\frak N_{\Gf (\tilde \GS )}(DC^-_{\tilde \GD }(\tilde \GS ))),\]

with respect to the chosen class of tubular neighborhoods in \(\R ^{n+1}\times U_K^{n+1}\) if follows that for \(0<t<e^{-\Gf (\tilde \GD )}\) the truncation \(b_t^\GG \) is \(\Ge \)-sufficient for \(b_t\) in some neighborhood of \(l^{-1}(\mathcal {E}_{t,\psi }(\GG ))\) with respect to the class of normal tubular neighborhoods.

\[b_{t,\GG }\in V\cap j_t^{-1}(DC^-_{\tilde \GD }(\tilde \GG )).\]

Then for \(\GG \) with \(\dim \GG =0\) put \(\GG _t=b_{t,\GG }\) and construct the others \(\GG _t\) inductively on dimension \(\dim \GG \): if \(\GG _t\) for \(\GG \) with \(\dim \GG <r\) have been constructed then for \(\GG \) with \(\dim \GG =r\) the cell \(\GG _t\) is the (open) cone on \(\bigcup _{\GS \in \mathcal {G}(\GG )\sminus \{\GG \}}\GS _t\) with the vertex \(b_{t,\GG }\). (This is the usual construction of dual partition turning in the case of triangulation to partition onto barycentric stars of simplices.)

By Theorem 6.7.A there exist \(t'_0\in (0,1)\) and function \(\psi :\mathcal {G}\to \R \) such that for any \(t\in (0,t'_0]\) and any face \(\GG \in \mathcal G\) the truncation \(b_t^\GG \) is \(\Ge \)-sufficient for \(b_t\) in some neighborhood of \(l^{-1}(\mathcal {E}_{t,\psi }(\GG ))\). Since cells \(\GG _t\) grow unboundedly when \(t\) runs to zero (if \(\dim \GG \ne 0\)) it follows that there exists \(t_0\in (0,t'_0]\) such that for \(t\in (0,t_0]\) for each face \(\GG \in \mathcal G\) the set \(\frak N_{\psi (\GG )}j^{-1}_t(DC^-_{\tilde \GD }(\GG ))\), and together with it the set \(\mathcal {E}_{t,\psi }(\GG )\), lie in the star of the cell \(\GG _t\), i.e. in \(\bigcup _{\GG \in \mathcal {G}(\GS )}\GS _t\). Let us show that for such \(t_0\) the conclusion of Theorem 4.3.A takes place.

Indeed, it follows from 6.7.A that there exists a homeomorphism \(h:\GG _t\times U_K^n\to \GG \times U_K^n\) with \(h(\GG _t\times y)=\GG \times y\) for \(y\in U_K^n\) such that \((\GG \times U_K^n,\) \(h(la(V_{K\Bbb R^n}(b_t))\cap \GG _t\times U_K^n))\) is \(K\)-chart of Laurent polynomial \(a^\GG \). Therefore the pair

\[(\cup _{\GG \in \mathcal {G}}\GG _t\times U_K^n,\, laV_{K\R ^n}(b_t)\cap (\cup _{\GG \in \mathcal {G}}\GG _t\times U_K^n))\]

is obtained in result of patchworking \(K\)-charts of Laurent polynomials \(a_1,\) …, \(a_s\). The function \(\Gf :\mathcal {G}(\GD )\to \R \), existing by Theorem 6.3.A applied to \(b_t\), can be chosen, as it follows from 6.4.A , in such a way that it should majorate any given in advance function \(\mathcal {G}(\GD )\to \R \). Choose \(\Gf \) in such a way that \(D_{\Gf ,\GD }(\GD )\supset \cup _{\GG \in \mathcal {G}}\GG _t\) and \(D_{\Gf ,\GD }(\GS )\cap \p D_{\Gf ,\GD }(\GD )\supset \mathcal {E}_{t,\psi }(\GS ) \cap \p D_{\Gf ,\GD }(\GD )\) for \(\GS \in \mathcal {G}(\GD )\sminus \{\GD \}\). As it follows from 6.7.A , there exists a homeomorphism

\begin{multline} (\bigcup _{\GG \in \mathcal {G}}\GG _t\times U_K^n,\,la V_{K\R ^n}(b_t)\cap (\bigcup _{\GG \in \mathcal {G}}\GG _t\times U_K^n))\to \\(D_{\Gf ,\GD }(\GD )\times U_K^n,\,la V_{K\R ^n}(b_t)\cap ( D_{\Gf ,\GD }(\GD )\times U_K^n)) \end{multline} turning \(\mathcal {E}_{t,\psi }(\GS )\cap \p (\bigcup _{\GG \in \mathcal {G}}\GG _t\times U_K^n)\) to \(\mathcal {E}_{t,\psi }(\GS )\cap \p D_{\Gf ,\GD }(\GD )\) for \(\GS \in \mathcal {G}(\GD )\sminus \{\GD \}\). Therefore \(K\)-chart of Laurent polynomial \(b_t\) is obtained by patchworking \(K\)-charts of Laurent polynomials \(a_1, \dots , a_s\).