[Paper Review] How many zeroes? Counting the number of solutions of systems of polynomials via geometry at infinity (Draft II)

Pinaki Mondal|arXiv (Cornell University)|Jun 14, 2018

Polynomial and algebraic computation41 references3 citations

TL;DR

This paper presents a toric geometry-based method to count the number of isolated solutions of n polynomial equations in n variables over an algebraically closed field, extending Bernstein’s theorem from the torus (k×)ⁿ to the full affine space kⁿ. The key contribution is a characterization of generic systems via Newton polytopes and a formula for solution counts using mixed volumes, resolving a long-standing problem in algebraic geometry with complete proofs over arbitrary algebraically closed fields.

ABSTRACT

In this book we describe an approach through toric geometry to the following problem: estimate the number (counted with appropriate multiplicity) of isolated solutions of n polynomial equations in n variables over an algebraically closed field k. The outcome of this approach is the number of solutions for systems in terms of their polytopes, and an explicit characterization of what makes a system generic. The pioneering work in this field was done in the 1970s by Kushnirenko, Bernstein and Khovanskii, who completely solved the problem of counting solutions of generic systems on the torus (k\0)^n. In the context of our problem, however, the natural domain of solutions is not the torus, but the affine space k^n. There were a number of works on extension of Bernstein's theorem to the case of affine space, and recently it has been completely resolved, the final steps having been carried out by the author. The aim of this book is to present these results in a coherent way. We start from the beginning, namely Bernstein's beautiful theorem which expresses the number of solutions of generic systems in terms of the mixed volume of their Newton polytopes. We give complete proofs, over arbitrary algebraically closed fields, of Bernstein's theorem and its recent extension to the affine space, and describe some open problems. We also apply the developed techniques to derive and generalize Kushnirenko's results on Milnor numbers of hypersurface singularities which in 1970s served as a precursor to the development of toric geometry. Care was taken to make this book as elementary as possible. In particular, we develop all the necessary algebraic geometry (modulo some explicitly stated basic results).

Motivation & Objective

• To provide a complete and coherent treatment of the extension of Bernstein’s theorem from the torus to affine space kⁿ.
• To characterize generic systems of polynomial equations in terms of their Newton polytopes and mixed volumes.
• To offer elementary, self-contained proofs of Bernstein’s theorem and its recent extension, accessible over arbitrary algebraically closed fields.
• To generalize Kushnirenko’s results on Milnor numbers of hypersurface singularities using the developed framework.
• To identify and discuss open problems in the enumeration of solutions of polynomial systems using geometric methods.

Proposed method

• Utilizes toric geometry to analyze the behavior of polynomial systems at infinity, particularly focusing on the geometry of Newton polytopes.
• Applies mixed volume theory to compute the number of isolated solutions of generic systems in affine space.
• Employs a systematic study of the compactification of the torus and its boundary divisors to understand solution counts beyond the torus.
• Develops foundational algebraic geometry concepts from scratch, ensuring accessibility while maintaining mathematical rigor.
• Uses the structure of Newton polytopes to define and characterize genericity conditions for polynomial systems.
• Establishes a precise correspondence between the mixed volume of Newton polytopes and the number of solutions, counted with multiplicity.

Experimental results

Research questions

• RQ1 How can Bernstein’s theorem on solution counting via mixed volumes be extended from the torus to the full affine space kⁿ?
• RQ2 What geometric conditions on Newton polytopes ensure that a system of polynomials has only finitely many isolated solutions in kⁿ?
• RQ3 How does the behavior of solutions at infinity affect the total count of solutions in affine space?
• RQ4 In what way do Newton polytopes and mixed volumes characterize genericity in polynomial systems over algebraically closed fields?
• RQ5 How can the results of Kushnirenko on Milnor numbers be generalized using the framework of toric geometry and solution counting?

Key findings

• The number of isolated solutions of a generic system of n polynomials in n variables over an algebraically closed field is given by the mixed volume of their Newton polytopes.
• The extension of Bernstein’s theorem to affine space is fully resolved, providing a complete formula for solution counts in kⁿ.
• Genericity is characterized geometrically by the relative position and structure of the Newton polytopes, ensuring transversality at infinity.
• The method yields a uniform treatment of solution counting that includes both toric and affine solutions, unifying previous results.
• The framework generalizes Kushnirenko’s work on Milnor numbers by linking them to mixed volume computations and solution counts.
• All results are proven over arbitrary algebraically closed fields, with full technical development provided in the book.