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Newton numbers, vanishing polytopes and algebraic degrees

Fedor Selyanin

TL;DR

This work develops a unified combinatorial framework for analyzing Newton numbers of convex polytopes in relation to polynomial hypersurfaces, introducing the ell- and e-Newton numbers to refine the classical Kouchnirenko theory. It proves a central B_k-Theorem: a convenient polytope has vanishing Newton number if and only if it is a Cayley (B_k) sum based on a coordinate subspace, using the Furukawa–Ito classification of dual defective sets. The ell-Newton number yields nonnegativity and thinness results, while the e-Newton number connects to zero-dimensional critical complete intersections and to a broad range of algebraic degrees (ML, MML, ED, polar), including applications to GKZ polytopes. The paper also develops a robust polyhedral refinement of Katz–Stapledon theory via strong formal subdivisions with boundary and agglutination, enabling a systematic treatment of local and global Newton-type invariants in cones. Overall, the results advance Arnold’s monotonicity problem, provide new tools for toric geometry, and unify several algebraic-degree formulas under the e-Newton framework.

Abstract

Consider a polynomial $f$ with a convenient Newton polytope $P$ and generic complex coefficients. By the global version of the Kouchnirenko formula, the hypersurface $\{f = 0\} \subset \mathbb{C}^n$ has the homotopy type of a bouquet of $(n-1)$-spheres, and the number of spheres is given by a certain alternating sum of volumes, called the Newton number $ν(P)$. Using the Furukawa-Ito classification of dual defective sets, we classify convenient Newton polytopes with vanishing Newton numbers as certain Cayley sums called $B_k$-polytopes. These $B_k$-polytopes generalize the $B_1$- and $B_2$-facets appearing in the local monodromy conjecture in the Newton non-degenerate case. Our classification provides a partial solution to Arnold's monotonicity problem. The local $h^*$-polynomial (or $\ell^*$-polynomial) is a natural invariant of lattice polytopes that refines the $h^*$-polynomial coming from Ehrhart theory. We obtain decomposition formulas for the Newton number, for instance, prove the inequality $ν(P) \ge \ell^*(P;1)$. The $B_k$-polytopes are non-trivial examples of thin polytopes. We generalize the Newton number in two independent ways: the $\ell$-Newton number and the $e$-Newton number. The $\ell$-Newton number comes from Ehrhart theory, namely, from certain generalizations of Katz-Stapledon decomposition formulas, and its properties are central to our proof that the $B_k$-polytopes are thin. The $e$-Newton number is the number of points of zero-dimensional critical complete intersections. Vanishing of the $e$-Newton number characterizes dual defective sets. Furthermore, the $e$-Newton number calculates algebraic degrees (such as Maximum Likelihood, Euclidean Distance and Polar degrees). For instance, we show that all known formulas for these algebraic degrees in the Newton non-degenerate case are implied by basic properties of the $e$-Newton number.

Newton numbers, vanishing polytopes and algebraic degrees

TL;DR

This work develops a unified combinatorial framework for analyzing Newton numbers of convex polytopes in relation to polynomial hypersurfaces, introducing the ell- and e-Newton numbers to refine the classical Kouchnirenko theory. It proves a central B_k-Theorem: a convenient polytope has vanishing Newton number if and only if it is a Cayley (B_k) sum based on a coordinate subspace, using the Furukawa–Ito classification of dual defective sets. The ell-Newton number yields nonnegativity and thinness results, while the e-Newton number connects to zero-dimensional critical complete intersections and to a broad range of algebraic degrees (ML, MML, ED, polar), including applications to GKZ polytopes. The paper also develops a robust polyhedral refinement of Katz–Stapledon theory via strong formal subdivisions with boundary and agglutination, enabling a systematic treatment of local and global Newton-type invariants in cones. Overall, the results advance Arnold’s monotonicity problem, provide new tools for toric geometry, and unify several algebraic-degree formulas under the e-Newton framework.

Abstract

Consider a polynomial with a convenient Newton polytope and generic complex coefficients. By the global version of the Kouchnirenko formula, the hypersurface has the homotopy type of a bouquet of -spheres, and the number of spheres is given by a certain alternating sum of volumes, called the Newton number . Using the Furukawa-Ito classification of dual defective sets, we classify convenient Newton polytopes with vanishing Newton numbers as certain Cayley sums called -polytopes. These -polytopes generalize the - and -facets appearing in the local monodromy conjecture in the Newton non-degenerate case. Our classification provides a partial solution to Arnold's monotonicity problem. The local -polynomial (or -polynomial) is a natural invariant of lattice polytopes that refines the -polynomial coming from Ehrhart theory. We obtain decomposition formulas for the Newton number, for instance, prove the inequality . The -polytopes are non-trivial examples of thin polytopes. We generalize the Newton number in two independent ways: the -Newton number and the -Newton number. The -Newton number comes from Ehrhart theory, namely, from certain generalizations of Katz-Stapledon decomposition formulas, and its properties are central to our proof that the -polytopes are thin. The -Newton number is the number of points of zero-dimensional critical complete intersections. Vanishing of the -Newton number characterizes dual defective sets. Furthermore, the -Newton number calculates algebraic degrees (such as Maximum Likelihood, Euclidean Distance and Polar degrees). For instance, we show that all known formulas for these algebraic degrees in the Newton non-degenerate case are implied by basic properties of the -Newton number.

Paper Structure

This paper contains 36 sections, 60 theorems, 95 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. $\mathbf{\ell}$-Newton number and thin polytopes
  3. $\mathbf{e}$-Newton number and Algebraic degrees
  4. Preliminaries: Katz-Stapledon decomposition formulas
  5. Eulerian posets, $\mathbf{h}$- and $\mathbf{g}$-polynomial
  6. Local $\mathbf{h}$-polynomials of strong formal subdivisions
  7. $\mathbf{h^*}$- and local $\mathbf{h^*}$-polynomials of lattice polytopes
  8. Strong formal subdivision with boundary
  9. Poset with boundary and twins-poset
  10. Strong formal subdivisions with boundaries
  11. Subdivisions of convenient polytopes in cones
  12. Generalization of $\mathbf{h^*}$- and $\mathbf{\ell^*}$-polynomial
  13. $\mathbf{h^*}$- and $\mathbf{\ell^*}$-polynomial of polyhedral strong formal subdivision
  14. $\mathbf{h^*}$- and $\ell^*$-polynomial of agglutination
  15. Manifestations of the Newton numbers
  16. ...and 21 more sections

Key Result

Theorem 1.3

A convenient polytope $P$ in the cone $\mathbb{R}^m_{\ge 0} \oplus \mathbb{R}^{n-m}$ is negligible (i.e. $\nu(P) = 0$) if and only if it is a $B_k$-polytope based on a coordinate subspace $\mathbb{R}^k_{\ge 0} \subset \mathbb{R}^m_{\ge 0}$.

Figures (7)

  • Figure 1: Hasse diagram of a poset (left) and the twins-poset (right) born from it
  • Figure 2: Convenient square $P = OABC$ in the cone $C = \mathbb{R}^2_{\ge 0}$ with red boundary $\partial_C (P)$
  • Figure 3: Strong formal subdivision (dashed vectors) between the face poset $\mathcal{P}$ of the square $OABC$ (right) and $\mathcal{C}^\pm$ (left). Hasse diagrams of the boundary poset $\mathcal{P}_B \subset \mathcal{P}$, the lower set $\mathcal{C}^- \subset \mathcal{C}^\pm$ and the mapping between them are drawn red.
  • Figure 4: The support $\mathbf P$ in dimension $4$
  • Figure 5: The polytope $P$ and the triangle $\Delta^\circ$ in the cone $C_d$
  • ...and 2 more figures

Theorems & Definitions (177)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: $B_k$-Theorem, see § \ref{['B_k_sec']}
  • Remark 1.4
  • Definition 1.6
  • Corollary 1.7
  • Definition 1.8
  • Corollary 1.11
  • Definition 2.1
  • Remark 2.2
  • ...and 167 more