Dimension Results for Extremal-Generic Polynomial Systems over Complete Toric Varieties

TL;DR

This work develops a dimension theory for varieties defined by extremal-generic polynomial systems in the Cox ring of a complete toric variety, using the orbit-cone correspondence to reduce dimension questions to torus-orbit intersections. It provides explicit combinatorial criteria for when such systems define (i) subschemes of expected dimension, (ii) complete intersections, and (iii) regular sequences, with detailed treatment for polytopal algebras and weighted homogeneous systems. A key result is a precise dimension formula dim Y = max_{σ: E_σ essential} (n − dim(σ) − |E_σ|) and a corresponding regular-sequence criterion |E_F| ≥ dim(F) + r − n, with extensions to weighted projective spaces and polytopal settings. The paper also proves NP-hardness for deciding generic regularity, highlighting fundamental computational limits in selecting toric compactifications for sparse elimination. These contributions impact sparse elimination theory and Gröbner-bases computation by enabling dimension-aware analysis and revealing intrinsic complexity barriers.

Abstract

We study polynomial systems with prescribed monomial supports in the Cox rings of toric varieties built from complete polyhedral fans. We present combinatorial formulas for the dimensions of their associated subvarieties under genericity assumptions on the coefficients of the polynomials. Using these formulas, we identify at which degrees generic systems in polytopal algebras form regular sequences. Our motivation comes from sparse elimination theory, where knowing the expected dimension of these subvarieties leads to specialized algorithms and to large speed-ups for solving sparse polynomial systems. As a special case, we classify the degrees at which regular sequences defined by weighted homogeneous polynomials can be found, answering an open question in the Gröbner bases literature. We also show that deciding whether a sparse system is generically a regular sequence in a polytopal algebra is hard from the point of view of theoretical computational complexity.

Figures (3)

• Figure 1: Two ways of homogenizing a imaginary genus-$2$ hyperelliptic equation $y^2=f(x)$. On the left, the projective homogenization in $\mathbb{P}^2$; on the right, the homogenization with respect to the weighted projective space, assigning weight $g+1$ on the variable $y$.
• Figure 2: This picture represents the polytope $P_{D, \{0\}}$ and the set $A_{D,\{0\}}$ for the effective divisor $D=D_{\rho_1}+D_{\rho_2}+3\,D_{\rho_3}+4\,D_{\rho_4}$ on the toric variety $X$ defined by the $2$-dimensional complete fan whose rays are $\rho_1=\mathbb{R}_>\cdot(0,1), \rho_2=\mathbb{R}_>\cdot(-1, -2), \rho_3=\mathbb{R}_>\cdot(1, -1), \rho_4=\mathbb{R}_>\cdot(2,1)$. This example shows that the convex hull of $A_{D,\{0\}}$ need not be equal to $P_{D,\{0\}}$.
• Figure 3: The two-dimensional fan associated to the Hirzebruch surface $\mathcal{H}_2$. On the left, the polytope $P_{D,\{0\}}$ associated to the divisor $D = D_1 + D_4$ where $D_1$ is the $T$-divisor of the ray $\mathbb{R}_{\geq 0}(1,0)$ and $D_4$ is the $T$-divisor of the ray $\mathbb{R}_{\geq 0}(0, -1)$. The lattice points of $P_{D,\{0\}}$ are in bijection with the monomials in the Cox ring $\mathbb{C}[x,y,z,t]$ of degree $(1,1)$, where $\deg(x^a y^bz^ct^d)=(c+a-2b, d+b)$. On the right, the Newton polytope of the polynomial in \ref{['example:Coxring']} inside $P_{D,\{0\}}$.

Theorems & Definitions (62)

• Example
• Example 1.1
• Lemma 1.2
• proof
• Lemma 1.3
• proof
• Proposition 1.4
• proof
• Proposition 1.5
• proof