Breaking the symmetry in excess intersection and counting solutions of systems of polynomials

2025/02/25 by Pinaki Mondal, Mondal, Pinaki
Computer Science · Mathematics · #14C17 #14M25 #14N10 #52B20 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #FOS: Mathematics #Mathematical functions and polynomials #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2502.18276

openalex publication_date 2025/02/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We revisit the fundamental problem of assigning intersection multiplicities to subsets of solutions of (square) systems of polynomials. Severi [Ann. Mat. Pura Appl. 26 (4), 1947] suggested an intuitive dynamic solution to this problem which was later corrected and made rigorous by Lazarsfeld [Compos. Math. 43, 1981]. We consider an asymmetric variant of this approach and find an explicit description of the resulting "ordered intersection multiplicity" which opens pathways to step by step solutions to the affine Bézout problem of counting isolated solutions to (square) systems of polynomials via "Bernstein-Kushnirenko type" estimates in terms of Newton diagrams. To illustrate our methods we compute the number of common tangent lines to 4 general spheres in the affine 3-space (which is known to be 12 due to Macdonald, Pach, and Theobald [Discrete Comput. Geom. 26 (1), 2001]) via certain ordered intersection multiplicities on the corresponding Grassmannian.

Related

• Intersection multiplicity, Milnor number and Bernstein's theorem
• Multiplicity estimates, analytic cycles and Newton polytopes
• Engineered complete intersections: slightly degenerate Bernstein--Kouchnirenko--Khovanskii
• On the multiplicity of solutions of a system of algebraic equations
• The sharp bound for the number of real solutions to polynomial equation systems