Singularities of square-free polynomials

TL;DR

The paper proves that hypersurfaces defined by irreducible square-free polynomials have rational singularities in characteristic zero, and it generalizes this to polynomials of the form $gL+h$ with $L$ linear and $deg(h)=deg(g)+1$, provided $g$ is irreducible and does not divide $h$. The authors develop minimal log discrepancy techniques, establishing bounds $mld_P(A^n,Z) \ge n - mult_P(Z)$ and $mult_W(Z) \le r-1$ for codimension $r\ge 2$, which imply $mld_{\eta_W}(A^n,Z) \ge 1$ and hence rational singularities; a homogenization step yields the homogeneous case. Consequently, the results unify and extend BW’s findings for matroid- and Feynman-diagram–related polynomials, and they apply to non-square-free instances under suitable setups. The work also discusses extensions to positive characteristic via $F$-rational singularities and clarifies limitations by showing that not all Lorentzian polynomials have rational or log canonical singularities, while noting alternative proofs via Brion’s multiplicity-free framework.

Abstract

We prove that hypersurfaces defined by irreducible square-free polynomials have rational singularities. As an easy consequence, we deduce that certain (possibly non-square-free) polynomials associated to pairs of square-free polynomials define hypersurfaces with rational singularities. This extends results on certain classes of polynomials associated to matroids and Feynman diagrams in [BW].

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 1.3
• Remark 2.1
• Theorem 2.2
• proof
• Remark 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof