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A finiteness result towards the Casas-Alvero Conjecture

Soham Ghosh

TL;DR

The paper proves a finiteness result toward the Casas-Alvero conjecture by bounding the dimension of the intersection of higher discriminant loci associated with monic univariate polynomials. It develops a framework of D-subschemes and linear reductions, then translates the conjecture into a complete-intersection problem for higher discriminant hypersurfaces and their Vieta-relations, culminating in a dimension bound of at most two for the relevant intersection. This yields that the arithmetic Casas-Alvero scheme $X_n$ has finite rational points over any field and, more generally, that there are only finitely many counterexamples up to affine changes in degree $n$. The work also introduces intermediate arithmetic Casas-Alberto schemes $X_n[j]$ and proves rigidity consequences and partial complete-intersection results, including a lower bound on how many of these intermediate schemes attain the complete-intersection property. Together, these results advance a structural, deformation-based approach to the Casas-Alvero conjecture, connecting algebraic geometry, discriminants, and arithmetic geometry in a way that provides both finiteness statements and finer geometric constraints.

Abstract

The Casas-Alvero conjecture predicts that every univariate polynomial over an algebraically closed field of characteristic zero sharing a common factor with each of its Hasse-Schmidt derivatives is a power of a linear polynomial. The conjecture for polynomials of a fixed degree is equivalent to the projective variety of such polynomials being one-dimensional. In this paper, we show that for any algebraically closed field of arbitrary characteristic, this variety is at most two-dimensional for all positive degrees. Consequently, we show that the associated arithmetic Casas-Alvero scheme in any positive degree has finitely many rational points over any field. Along the way, we prove several rigidity results towards the conjecture. We also introduce intermediate arithmetic Casas-Alvero schemes and show that their $\mathbb{K}$ points form an almost complete intersection over any algebraically closed field $\mathbb{K}$. Furthermore, we consider the question of when they form a complete intersection.

A finiteness result towards the Casas-Alvero Conjecture

TL;DR

The paper proves a finiteness result toward the Casas-Alvero conjecture by bounding the dimension of the intersection of higher discriminant loci associated with monic univariate polynomials. It develops a framework of D-subschemes and linear reductions, then translates the conjecture into a complete-intersection problem for higher discriminant hypersurfaces and their Vieta-relations, culminating in a dimension bound of at most two for the relevant intersection. This yields that the arithmetic Casas-Alvero scheme has finite rational points over any field and, more generally, that there are only finitely many counterexamples up to affine changes in degree . The work also introduces intermediate arithmetic Casas-Alberto schemes and proves rigidity consequences and partial complete-intersection results, including a lower bound on how many of these intermediate schemes attain the complete-intersection property. Together, these results advance a structural, deformation-based approach to the Casas-Alvero conjecture, connecting algebraic geometry, discriminants, and arithmetic geometry in a way that provides both finiteness statements and finer geometric constraints.

Abstract

The Casas-Alvero conjecture predicts that every univariate polynomial over an algebraically closed field of characteristic zero sharing a common factor with each of its Hasse-Schmidt derivatives is a power of a linear polynomial. The conjecture for polynomials of a fixed degree is equivalent to the projective variety of such polynomials being one-dimensional. In this paper, we show that for any algebraically closed field of arbitrary characteristic, this variety is at most two-dimensional for all positive degrees. Consequently, we show that the associated arithmetic Casas-Alvero scheme in any positive degree has finitely many rational points over any field. Along the way, we prove several rigidity results towards the conjecture. We also introduce intermediate arithmetic Casas-Alvero schemes and show that their points form an almost complete intersection over any algebraically closed field . Furthermore, we consider the question of when they form a complete intersection.
Paper Structure (20 sections, 32 theorems, 83 equations)

This paper contains 20 sections, 32 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Aim and main results of the paper
  3. Existing results and methods
  4. Organization of the paper
  5. Global notations and Definitions
  6. D-subschemes and monomial affine $\mathbb{K}$-schemes
  7. Preliminaries on D-subschemes
  8. Linear reduction of affine $\mathbb{K}$-schemes
  9. D-subschemes of affine schemes over positive characteristic
  10. D-subschemes of prinicpal monomial affine $\mathbb{K}$-schemes
  11. Geometry of higher discriminants
  12. Geometry of monic univariate polynomials of degree $n$
  13. Higher discriminant hypersurfaces
  14. Arithmetic Casas-Alvero scheme and higher discriminants
  15. Proof of the main results
  16. ...and 5 more sections

Key Result

Theorem A

For $n\geq 3$ and any algebraically closed field $\mathbb{K}$, $\dim \bigcap_{i=1}^{n-1}X^{i}_n(\mathbb{K}) \leq 2$.

Theorems & Definitions (80)

  • Conjecture CA: Casas-Alvero, 2001
  • Theorem A: =Theorem \ref{['PropNew']}
  • Theorem B: =Corollary \ref{['maincor1']}
  • Corollary C: =Corollary \ref{['maincor2']}
  • Theorem D: =Theorem \ref{['thmsingular']}
  • Theorem E: =Corollary \ref{['cor:jC']}
  • Proposition
  • Proposition
  • Definition 3.1: D-ideal and D-subscheme
  • Remark 3.2
  • ...and 70 more