[Paper Review] Elimination theory and Newton polytopes

Askold Khovanskiĭ, Alexander Esterov|arXiv (Cornell University)|Nov 4, 2006

Polynomial and algebraic computation5 references4 citations

TL;DR

This paper develops a comprehensive framework for elimination theory using Newton polytopes, showing how the Newton polytope and leading coefficients of the defining equation of a projected complete intersection can be computed from the Newton polytopes and leading coefficients of the original equations. The key contribution is identifying the Newton polytope of the projection as a mixed fiber polytope, up to translation and scaling, and providing explicit formulas for leading coefficients under general position conditions.

ABSTRACT

We study elimination theory in the context of Newton polytopes and develop its convex-geometric counterpart.

Motivation & Objective

• To describe the Newton polytope and leading coefficients of the defining equation of a projected complete intersection in terms of the Newton polytopes and leading coefficients of the original equations.
• To establish that the Newton polytope of the projection is a mixed fiber polytope, up to translation and scaling, of the original Newton polytopes.
• To provide explicit formulas for the leading coefficients of the projected equation under general position conditions on the Newton polytopes.
• To unify and generalize classical results such as Kouchnirenko-Bernstein’s formula, Khovanskii’s product formula, and Gelfand-Khovanskii’s sum formula within a single elimination-theoretic framework.
• To extend the theory to germs of analytic functions and unbounded polyhedra using Minkowski volume constructions and stabilization techniques.

Proposed method

• Define the projection of a complete intersection via a Laurent polynomial equation $ g = 0 $, where $ g $ is derived from the original equations $ f_1 = \cdots = f_k = 0 $ with given Newton polytopes.
• Introduce the concept of the mixed fiber polytope as the key geometric object that captures the Newton polytope of the projected variety.
• Prove that the mixed fiber polytope is an increasing function of the input polytopes $ \Delta_1, \ldots, \Delta_k $, and is equal to the mixed fiber polytope up to translation and scaling.
• Use Minkowski summation and mixed volume theory for unbounded polyhedra parallel to the positive orthant, employing $ M $-far stabilizations to define well-defined mixed volumes.
• Apply a generalized Bernstein theorem for germs of analytic functions, linking the isolated multiplicity of common roots to the mixed volume of Newton polyhedra.
• Derive explicit formulas for leading coefficients of $ g $ using symmetric functions and the values of monomials over common roots, under general position assumptions on the faces of the Newton polytopes.

Experimental results

Research questions

• RQ1 How can the Newton polytope of the projection of a complete intersection be characterized in terms of the Newton polytopes of the defining equations?
• RQ2 What conditions on the Newton polytopes ensure that the leading coefficients of the projected equation can be computed explicitly?
• RQ3 How does the mixed fiber polytope relate to the Newton polytope of the projected variety, and what is its geometric and algebraic significance?
• RQ4 Under what conditions is the mixed volume of unbounded polyhedra (parallel to the positive orthant) well-defined, and how does this relate to the multiplicity of isolated common roots?
• RQ5 Can classical formulas in algebraic geometry—such as Kouchnirenko-Bernstein, Khovanskii’s product, and Gelfand-Khovanskii’s sum—be derived as special cases of a unified elimination-theoretic framework?

Key findings

• The Newton polytope of the projection of a complete intersection is the mixed fiber polytope of the Newton polytopes of the defining equations, up to translation and scaling.
• The mixed fiber polytope is an increasing function of the input polytopes $ \Delta_1, \ldots, \Delta_k $, ensuring monotonicity in the elimination process.
• Under general position conditions (Definition 5.2), the coefficients of monomials corresponding to vertices of the Newton polytope of $ g $ can be computed explicitly using Theorems 4.6 and 5.13.
• The number of common roots of $ n $ Laurent polynomials with given Newton polytopes and generic coefficients is given by Kouchnirenko-Bernstein’s formula, which is recovered as a special case.
• The product of the values of a monomial over the common roots is given by Khovanskii’s product formula, which is derived as a coefficient computation in the elimination framework.
• For germs of analytic functions with isolated common roots, the multiplicity $ \mu $ satisfies $ \mu \geq n!V $, where $ V $ is the mixed volume of their Newton polyhedra, and equality holds under general position conditions on the faces of the polyhedra.