Shifted Partial Derivative Polynomial Rank and Codimension

TL;DR

This paper introduces the SPDP framework, a linear-algebraic formalism for shifted partial derivatives, providing new tools for circuit lower bounds and analyzing polynomial complexity.

Contribution

The paper develops the SPDP matrix formalism, establishing structural properties and bounds, and applies it to circuit complexity analysis with a focus on codimension and rank.

Findings

SPDP matrices are invariant under variable symmetries.

Monotonicity properties of SPDP rank and codimension.

Width-to-rank bounds for local circuit models.

Abstract

Shifted partial derivative (SPD) methods are a central algebraic tool for circuit lower bounds, measuring the dimension of spaces of shifted derivatives of a polynomial. We develop the Shifted Partial Derivative Polynomial (SPDP) framework, packaging SPD into an explicit coefficient-matrix formalism. This turns shifted-derivative spans into concrete linear-algebraic objects and yields two dual measures: SPDP rank and SPDP codimension. We define the SPDP generating family, its span, and the SPDP matrix M_{k,l}(p) inside a fixed ambient coefficient space determined by the (k,l) regime, so rank is canonical and codimension is a deficit from ambient fullness. We prove structural properties needed for reuse: monotonicity in the shift/derivative parameters (with careful scoping for |S|=k versus |S|<=k conventions), invariance under admissible variable symmetries and basis changes, and robustness across standard Boolean/multilinear embeddings. We then give generic width-to-rank upper-bound templates for local circuit models via profile counting, separating the model-agnostic SPDP toolkit from additional compiled refinements used elsewhere. We illustrate the codimension viewpoint on representative examples.