×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite