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Proof of the Casas-Alvero conjecture

Soham Ghosh

TL;DR

The paper resolves the Casas-Alvero conjecture for all degrees $d \ge 3$ over characteristic zero fields by translating the problem into regularity of certain homogeneous polynomials and proving this regularity via Koszul homology. The core method employs an induction on the degree, rephrasing the conjecture as a complete intersection problem and analyzing a truncated Koszul complex with a carefully constructed filtration. A crucial ingredient is establishing the injectivity of a multiplication map on zeroth Koszul homology, which hinges on the ground field having characteristic zero. The result unifies prior partial progress, yields finiteness conclusions for arithmetic Casas-Alvero schemes, and provides a new algebraic proof strategy that avoids characteristic-dependent obstructions.

Abstract

The Casas-Alvero conjecture states that if $f(X)$ is a monic univariate polynomial of degree $d$ over a characteristic $0$ field $\mathbb{K}$ such that $\gcd(f, f_{i})$ is non-trivial for each $i=1, \dots, d-1$, then $f(X)=(X-α)^d$ for some $α\in \mathbb{K}$. In this paper, we prove the Casas-Alvero conjecture for polynomials of any degree $d\geq 3$ over any characteristic $0$ field using Koszul homology.

Proof of the Casas-Alvero conjecture

TL;DR

The paper resolves the Casas-Alvero conjecture for all degrees over characteristic zero fields by translating the problem into regularity of certain homogeneous polynomials and proving this regularity via Koszul homology. The core method employs an induction on the degree, rephrasing the conjecture as a complete intersection problem and analyzing a truncated Koszul complex with a carefully constructed filtration. A crucial ingredient is establishing the injectivity of a multiplication map on zeroth Koszul homology, which hinges on the ground field having characteristic zero. The result unifies prior partial progress, yields finiteness conclusions for arithmetic Casas-Alvero schemes, and provides a new algebraic proof strategy that avoids characteristic-dependent obstructions.

Abstract

The Casas-Alvero conjecture states that if is a monic univariate polynomial of degree over a characteristic field such that is non-trivial for each , then for some . In this paper, we prove the Casas-Alvero conjecture for polynomials of any degree over any characteristic field using Koszul homology.
Paper Structure (14 sections, 5 theorems, 12 equations)

This paper contains 14 sections, 5 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Aim and main results of the paper
  3. Idea of the proof
  4. Existing results and methods
  5. Historical note
  6. Proof of Theorem \ref{['MainTheorem']}
  7. Notations and Setup of the proof of Theorem \ref{['MainTheorem']}
  8. The hypothesis
  9. The inductive step
  10. Proof of Theorem \ref{['MainTheorem']}
  11. Filtration of the truncated Koszul complex $\widehat{\operatorname{K}}^n(j_1,\dots, j_{n-1})_{\bullet}$
  12. Homology computations
  13. Regularity of $\Phi^{\#}_{j_n}(HD^{n-1}_n\mathbf{x}_n)$ in $H_0(\widehat{\operatorname{K}}^{n}_{\bullet})$
  14. Injectivity of $\mu_{n,\star}: H_0(\widehat{\operatorname{K}}^{n}_{1,\bullet})\rightarrow H_0(\widehat{\operatorname{K}}^{n}_{2,\bullet})$

Key Result

Theorem A

Let $f(X)$ be a monic univariate polynomial of degree $d\geq 3$ over a characteristic zero field $\mathbb{K}$. Then $\gcd(f, f_{i})$ is non-trivial for each $i=1, \dots, d-1$ if and only if $f(X)=(X-\alpha)^d$ for some $\alpha\in \mathbb{K}$.

Theorems & Definitions (8)

  • Conjecture CA: Casas-Alvero, 2001
  • Theorem A
  • Corollary B
  • Corollary C
  • Proposition D
  • Proposition A: SGProposition $6.5$
  • Definition B
  • Definition C