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Discriminants of polynomials in several variables and triangulations of Newton polyhedra. (English. Russian original) Zbl 0741.14033
Leningr. Math. J. 2, No. 3, 449-505 (1991); translation from Algebra Anal. 2, No. 3, 1-62 (1990).
Let $$A\subset\mathbb{Z}^{n-1}$$ be a finite subset, $$\mathbb{C}^ A$$ the linear $$\mathbb{C}$$-space of Laurent polynomials $$f=\sum_{\omega\in A}a_ \omega X^ \omega$$, $$a_ \omega\in\mathbb{C}$$ in some indeterminates $$X$$ and $$\nabla_ 0\subset\mathbb{C}^ A$$ the set of those $$f$$ for which there is $$\kappa\in(\mathbb{C}^*)^{n-1}$$ such that $$f(\kappa)=(\partial f/\partial X_ i)(\kappa)=0$$ for all $$i$$. The closure $$\nabla_ A$$ of $$\nabla_ 0$$ is an irreducible variety defined in fact on $$\mathbb{Z}$$. When $$\nabla_ A$$ has codimension 1 then an irreducible polynomial $$\Delta_ A\in\mathbb{Z}[a_ \omega;\omega\in A]$$, which is zero on $$\nabla_ A$$, is unique up to the sign and it is called the $$A$$-discriminant. If $$\text{codim}(\nabla_ A)>1$$ then put $$\Delta_ A=1$$. The $$A$$- discriminant is homogeneous and satisfies the following quasi homogeneous $$(n-1)$$-conditions: “$$\sum_{\omega\in A}m(\omega)\cdot\omega\in\mathbb{Z}^{n-1}$$ is constant for all monomials $$\prod_{\omega\in A}a_ \omega^{m(\omega)}$$ which enter in $$\Delta_ A$$”. This notion extends the classical notions of discriminant and resultant. — Let $$A=\{\omega_ 1,\ldots,\omega_ N\}$$ and $$Y_ A$$ be the closure of the set $$\{(\kappa^{\omega_ 1},\ldots,\kappa^{\omega_ N}\mid\kappa\in\mathbb{C}^{*n-1}\}$$ in $$\mathbb{P}^{N-1}$$. Then $$\nabla_ A$$ and $$Y_ A$$ are dual projective varieties and the description of $$\Delta_ A$$ follows if we can describe the equations of the dual projective variety of a given projective one $$Y\subset\mathbb{P}^{N-1}$$ [see the authors’ previous paper in Sov. Math., Dokl. 39, No. 2, 385-389 (1989); translation from Dokl. Akad. Nauk SSSR 305, No. 6, 1294-1298 (1989; Zbl 0715.14042)].
Let $$G$$ be a free abelian group of rank $$n$$, $$G_ \mathbb{C}:=\mathbb{C}\otimes_ \mathbb{Z} G$$, $$\lambda:G\to(\mathbb{Q},+)$$ a nonzero group morphism, $$S\subset G$$ a finitely generated semigroup such that $$o\in S$$ and $$\lambda(s)\geq 1$$ for all $$s\in S$$, $$S_ e=\{t\in S\mid\lambda(t)=e\}$$ for $$e\in\mathbb{Q}$$ and $$A\subset S_ 1$$ a finite subset generating in $$G_ \mathbb{R}=\mathbb{R}\otimes_ \mathbb{Z} G$$ the same convex cone as $$S$$. For $$k\in\mathbb{Z}_ +$$, $$e\in\mathbb{Q}$$, $$\omega\in A$$ let $$\bigwedge^ k(e)$$ be the space of all maps $$S_{k+e}\to\bigwedge^ kG_ \mathbb{C}$$, $$\partial_ \omega:\bigwedge^ k(e)\to\bigwedge^{k+1}(e)$$ the map given by $$\partial_ \omega(\gamma)(u)=\omega\wedge\gamma(u-\omega)$$, if $$u-\omega\in S_{k+e}$$, otherwise $$\partial_ \omega(\gamma)(u)=0$$ and $$\partial_ f=\sum_{\omega\in A}a_ \omega\partial_ \omega$$ if $$f=\sum a_ \omega X^ \omega\in\mathbb{C}^ A$$. The complex $$(\overset{.}\bigwedge (e),\partial_ f)$$ is called the Cayley-Koszul complex. — Choose a basis $$u$$ in terms of $$\overset{.}\bigwedge(e)$$ and let $$E_ e(f)$$ be the determinant of the complex $$(\overset{.}\bigwedge (e),\partial_ f)$$ with respect to $$u$$ [see F. Fischer, Math. Z. 26, 497-550 (1927) or J.-K. Bismut and D. S. Freed, Commun. Math. Phys. 106, 159-176 (1986; Zbl 0657.58037)]. For $$e$$ sufficiently high $$E_ A(f):=E_ e(f)$$ is a polynomial of $$(a_ \omega)$$, $$f=\sum a_ \omega X^ \omega$$ which depends on $$e$$ only by a constant multiple. — Let $$Q_ A$$ be the convex closure of $$A$$ in $$G_ \mathbb{R}$$. If $$f=\sum a_ \omega X^ \omega$$ we can express $$E_ A(f)=\sum_ \varphi c_ \varphi\prod_{\omega\in A}a_ \omega^{\varphi(\omega)}$$, where $$\varphi$$ runs in the set $$\mathbb{Z}^ A_ +$$ of the maps $$A\to\mathbb{Z}_ +$$. Let $$M(E_ A)\subset\mathbb{R}^ A$$ be the convex closure of those $$\varphi\in\mathbb{Z}^ A_ +$$ for which $$c_ \varphi\neq 0$$. Then there exists a nice correspondence between the vertices of $$M(E_ A)$$ and some special triangulations of $$Q_ A$$.
The theory is applied to the following examples: the discriminant of a polynomial in two indeterminates, the resultant of two quadratric polynomials, the elliptic curve in Tate normal form…

##### MSC:
 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 13D25 Complexes (MSC2000) 14M12 Determinantal varieties