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On the gradient of the coefficient of the characteristic polynomial

Christian Ikenmeyer

TL;DR

This work introduces the bivariate Cayley–Hamilton theorem, unifying the classical Cayley–Hamilton theorem, Girard–Newton identities, and polynomially bounded algebraic branching programs for the determinant and the coefficients χ_{n,d} of the characteristic polynomial. The central result expresses the transpose of the gradient of χ_{n,d+1} as $$( abla \, χ_{n,d+1})^T = \sum_{i=0}^d (-1)^i \, χ_{n,d-i} \, X_n^i,$$ generalizing the adjugate and yielding corollaries such as CH, matrix Girard–Newton identities, and poly‑width ABPs over arbitrary commutative rings. The authors construct an ABP computing all χ_{n,d}$ that is a factor of 3 smaller in size and a factor of 2 narrower than the previous best by Mahajan–Vinay, avoiding clow sequences and instead using first‑order partial derivatives of χ_{n,d} in a CCP‑based combinatorial framework. The results hold in full generality over commutative rings and illuminate the interplay between bivariate complexity (width vs degree) and linear‑algebraic identities, with implications for ABP lower/upper bounds and geometric complexity theory.

Abstract

We prove the bivariate Cayley-Hamilton theorem, a powerful generalization of the classical Cayley-Hamilton theorem. The bivariate Cayley-Hamilton theorem has three direct corollaries that are usually proved independently: The classical Cayley-Hamilton theorem, the Girard-Newton identities, and the fact that the determinant and every coefficient of the characteristic polynomial has polynomially sized algebraic branching programs (ABPs) over arbitrary commutative rings. This last fact could so far only be obtained from separate constructions, and now we get it as a direct consequence of this much more general statement. The statement of the bivariate Cayley-Hamilton theorem involves the gradient of the coefficient of the characteristic polynomial, which is a generalization of the adjugate matrix. Analyzing this gradient, we obtain another new ABP for the determinant and every coefficient of the characteristic polynomial. This ABP has one third the size and half the width compared to the current record-holder ABP constructed by Mahajan-Vinay in 1997. This is the first improvement on this problem for 28 years. Our ABP is built around algebraic identities involving the first order partial derivatives of the coefficients of the characteristic polynomial, and does not use the ad-hoc combinatorial concept of clow sequences. This answers the 26-year-old open question by Mahajan-Vinay from 1999 about the necessity of clow sequences. We prove all results in a combinatorial way that on a first sight looks similar to Mahajan-Vinay, but it is closer to Straubing's and Zeilberger's constructions.

On the gradient of the coefficient of the characteristic polynomial

TL;DR

This work introduces the bivariate Cayley–Hamilton theorem, unifying the classical Cayley–Hamilton theorem, Girard–Newton identities, and polynomially bounded algebraic branching programs for the determinant and the coefficients χ_{n,d} of the characteristic polynomial. The central result expresses the transpose of the gradient of χ_{n,d+1} as generalizing the adjugate and yielding corollaries such as CH, matrix Girard–Newton identities, and poly‑width ABPs over arbitrary commutative rings. The authors construct an ABP computing all χ_{n,d}$ that is a factor of 3 smaller in size and a factor of 2 narrower than the previous best by Mahajan–Vinay, avoiding clow sequences and instead using first‑order partial derivatives of χ_{n,d} in a CCP‑based combinatorial framework. The results hold in full generality over commutative rings and illuminate the interplay between bivariate complexity (width vs degree) and linear‑algebraic identities, with implications for ABP lower/upper bounds and geometric complexity theory.

Abstract

We prove the bivariate Cayley-Hamilton theorem, a powerful generalization of the classical Cayley-Hamilton theorem. The bivariate Cayley-Hamilton theorem has three direct corollaries that are usually proved independently: The classical Cayley-Hamilton theorem, the Girard-Newton identities, and the fact that the determinant and every coefficient of the characteristic polynomial has polynomially sized algebraic branching programs (ABPs) over arbitrary commutative rings. This last fact could so far only be obtained from separate constructions, and now we get it as a direct consequence of this much more general statement. The statement of the bivariate Cayley-Hamilton theorem involves the gradient of the coefficient of the characteristic polynomial, which is a generalization of the adjugate matrix. Analyzing this gradient, we obtain another new ABP for the determinant and every coefficient of the characteristic polynomial. This ABP has one third the size and half the width compared to the current record-holder ABP constructed by Mahajan-Vinay in 1997. This is the first improvement on this problem for 28 years. Our ABP is built around algebraic identities involving the first order partial derivatives of the coefficients of the characteristic polynomial, and does not use the ad-hoc combinatorial concept of clow sequences. This answers the 26-year-old open question by Mahajan-Vinay from 1999 about the necessity of clow sequences. We prove all results in a combinatorial way that on a first sight looks similar to Mahajan-Vinay, but it is closer to Straubing's and Zeilberger's constructions.

Paper Structure

This paper contains 25 sections, 13 theorems, 43 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Commutative rings
  3. Motivation: Width as the bivariate perspective on ABPs
  4. Motivation: The bivariate viewpoint in complexity theory
  5. Main results: The bivariate viewpoint in linear algebra
  6. Corollaries
  7. Related work
  8. Girard--Newton, Cayley--Hamilton, and combinatorial proofs in algebra
  9. ABPs, GapL, and clow sequences
  10. Structure, homogenization, and geometric complexity theory
  11. Preliminaries
  12. Clow sequences
  13. The Bivariate Cayley--Hamilton Theorem
  14. The entries of the gradient
  15. Appendix
  16. ...and 10 more sections

Key Result

Corollary 2.1

Figures (7)

  • Figure 1: Direct implications between the theorems. The dashed implication only works over algebraically closed fields: one gets the matrix Girard--Newton identities from the classical Girard--Newton identities, and from there one derives $w(\chi_{n,d})\in\textup{poly}(n,d)$, but only over fields of characteristic zero. To obtain $w(\chi_{n,d})\in\textup{poly}(n,d)$ over arbitrary commutative rings, one either uses our bivariate Cayley--Hamilton theorem, or one has to use an entirely separate construction such as Tod92Val92MV97, see §\ref{['sec:relwork']}.
  • Figure 2: Over fields of characteristic 0. An ABP computing along the way all $\chi_{n,d}$ for $d\leq 4$, where any $n\in\mathbb{N}$ is fixed. Each edge represents a sub-ABP whose source is some $v_i$ and whose sink is some $v_j$, $i<j$.
  • Figure 3: An ABP computing along the way all $\chi_{n,d}$ for $d\leq n\leq 3$. Each non-constant edge represents a sub-ABP.
  • Figure 4: The situation in the proof of Theorem \ref{['thm:CayleyHamilton']}. The left and right subfigure are partner summands in the sign-inverting involution. The figures are adaptions of figures in Str83.
  • Figure 5: The situation in the proof of Theorem \ref{['thm:wdetnsquare']}. The left and right subfigure are partner summands in the sign-inverting involution.
  • ...and 2 more figures

Theorems & Definitions (27)

  • Corollary 2.1: The Cayley--Hamilton theorem
  • proof
  • Corollary 2.2: The adjugate matrix in the Fadeev--LeVerrier algorithm
  • proof
  • Corollary 2.3: The trace Cayley--Hamilton theorem, also known as the matrix Girard--Newton identities
  • proof
  • Corollary 2.4: The Girard--Newton identities
  • proof
  • Corollary 2.5
  • proof
  • ...and 17 more