Proof of the Casas-Alvero conjecture (2501.09272v1)

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 $f(X)=(X-\alpha)^d$ for some $\alpha\in \mathbb{K}$. 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.

• The paper proves that any monic polynomial of degree d≥3 in characteristic zero with non-trivial gcds with its derivatives must be a pure power (X−a)^d.
• It employs an induction-based approach along with Koszul homology to establish regular sequences in multivariate polynomial rings.
• The derived corollaries on 'bad primes' offer deeper insights into polynomial factorization, paving the way for further algebraic research.

Proof of the Casas-Alvero Conjecture

The paper by Soham Ghosh addresses the long-standing Casas-Alvero Conjecture, a problem in algebraic geometry that has intrigued mathematicians since it was first posited. The conjecture, originally formulated by E. Casas-Alvero in 2001, concerns monic univariate polynomials over fields with characteristic zero and the relationship between such polynomials and their derivatives. Specifically, the conjecture states that if for a monic polynomial $f(X) \in K[X]$ of degree $d$, the greatest common divisor (gcd) of $f$ and its derivative $f^{(i)}$ is non-trivial for each $i = 1, \ldots, d-1$, then $f(X)$ must be of the form $(X - a)^d$ for some $a \in K$.

Ghosh successfully proves the conjecture for all polynomials of degree $d \geq 3$ over any characteristic zero field, employing Koszul homology as a central tool. This marks significant progress as the conjecture was known to be false in fields of positive characteristic for degrees $d \geq 3$. The proof is constructed on an induction-based approach, reformulating the Casas-Alvero Conjecture as a problem concerning regular sequences in multivariate polynomial rings.

The primary result, stated as Theorem A, confirms that over characteristic zero fields, the conjecture holds true for degrees $d \geq 3$. The author also derives Corollaries B and C, further elucidating conditions under which the conjecture holds across different fields, involving a finite set of "bad primes" that determine if the conjecture's conditions are satisfied for a given polynomial degree.

The proof innovatively uses Koszul complexes to reduce the conjecture to proving injectivity on zeroth homology, a substantial departure from previous computational methods and theoretical approaches. The use of induction is pivotal, as each inductive step relies on proving that certain sequences of homogeneous polynomials form regular sequences when extended by one term, a step that crucially leverages the characteristic zero condition.

In the broader theoretical context, this paper not only contributes a definitive answer to the Casas-Alvero Conjecture in characteristic zero fields but also provides methods potentially applicable to other problems involving derivatives and polynomial factorization. Future research may explore extensions of these methods to different algebraic settings or delve into computational techniques to verify the conjecture in remaining open cases.

Overall, this work is a substantial addition to the field of polynomial algebra, solving a conjecture that has stimulated considerable inquiry and debate among mathematicians. By introducing novel approaches and rigorously proving the conjecture's validity under specified conditions, Ghosh sets a robust foundation for further paper and application in higher-order algebraic theory.