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A short proof for the parameter continuation theorem

Viktoriia Borovik, Paul Breiding

TL;DR

The paper addresses the Parameter Continuation problem underlying polynomial homotopy continuation and presents a short, algebraic proof using Gröbner bases instead of holomorphic bundle theory. It introduces the saturation ideal $I:J^\infty$ and a key lemma ensuring compatibility under specialization for general parameters, hence a fixed finite regular-root count $N$ outside a discriminant $Δ$. The contributions include a projection-degree interpretation of $N$, a practical discriminant computation method via a Gröbner basis of $I + \{1 - y h\}$, and illustrative examples that compute discriminants for parameterized families. The approach yields both theoretical insight into generic root counts and an explicit algorithm for discriminant computation, with implications for fundamental root-count results and related theorems in algebraic geometry.

Abstract

The Parameter Continuation Theorem is the theoretical foundation for polynomial homotopy continuation, which is one of the main tools in computational algebraic geometry. In this note, we give a short proof using Gröbner bases. Our approach gives a method for computing discriminants.

A short proof for the parameter continuation theorem

TL;DR

The paper addresses the Parameter Continuation problem underlying polynomial homotopy continuation and presents a short, algebraic proof using Gröbner bases instead of holomorphic bundle theory. It introduces the saturation ideal and a key lemma ensuring compatibility under specialization for general parameters, hence a fixed finite regular-root count outside a discriminant . The contributions include a projection-degree interpretation of , a practical discriminant computation method via a Gröbner basis of , and illustrative examples that compute discriminants for parameterized families. The approach yields both theoretical insight into generic root counts and an explicit algorithm for discriminant computation, with implications for fundamental root-count results and related theorems in algebraic geometry.

Abstract

The Parameter Continuation Theorem is the theoretical foundation for polynomial homotopy continuation, which is one of the main tools in computational algebraic geometry. In this note, we give a short proof using Gröbner bases. Our approach gives a method for computing discriminants.
Paper Structure (2 sections, 4 theorems, 20 equations)

This paper contains 2 sections, 4 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Gröbner bases, Saturation and Parameterized Ideals

Key Result

Theorem 1.1

Let $\mathcal{F}$ be a family of polynomial systems that consists of systems of $n$ polynomials $F(\mathbf x;\mathbf p)$ in $n$ variables $\mathbf x$ depending on $k$ parameters $\mathbf p$. For $\mathbf q\in\mathbb C^k$ denote Let $N:=\sup_{\mathbf q\in\mathbb C^k} N(\mathbf q)$. Then, $N<\infty$ and there exists a proper algebraic subvariety $\Delta \subsetneq \mathbb C^k$, called a discriminan

Theorems & Definitions (13)

  • Theorem 1.1: The Parameter Continuation Theorem
  • Example
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Example
  • ...and 3 more