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Partial Derivative Complexity of a Product of Linearly Independent Quadratics

Authors Nir Shalmon , Amir Shpilka

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LIPIcs.ICALP.2026.152.pdf
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ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • algebraic complexity theory
  • partial derivatives
  • arithmetic circuits
  • quadratic polynomials

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Abstract

The partial derivative method is a central tool in algebraic complexity, underlying lower bounds for multilinear formulas, bounded depth circuits, and algebraic branching programs. A key feature of this measure is its subadditivity and submultiplicativity, which are usually used to upper bound the measure. However, proving lower bounds requires bounding the measure of explicit polynomials from below, and in some cases, a sharp estimate is required. For example, a frequently used fact is that the dimension of the space spanned by order k partial derivatives of a product of n linearly independent linear functions is binom(n,k).
Beyond the linear case, however, not much is known about the behavior of the (general) partial derivative measure under multiplication. In particular, it has been conjectured that for algebraically independent polynomials g₁,… ,g_r ∈ ℂ[𝐱], the partial derivative complexity of the product ∏_{i=1}^r g_i(𝐱) grows exponentially with r (see [Chaugule et al., 2023]), but prior to this work such bounds were only known when the g_i’s are linear polynomials, or satisfy additional restrictions.
In this paper, we show a lower bound of exp(Ω(r^{1/6})) for the measure of a product of r linearly independent quadratic polynomials. This is the first result to show such a lower bound on the partial derivative measure of a product of nonlinear polynomials, without any further restrictions. Interestingly, we only assume linear independence, which is weaker than algebraic independence. Our proof relies on algebraic-geometric and combinatorial techniques, combining the Jacobian approach of [Chaugule et al., 2023] together with the theory of wide algebras introduced in [Ananyan and Hochster, 2020; Oliveira and Sengupta, 2022; Garg et al., 2023]. To our knowledge, this is the first use of wide-algebra techniques for proving lower bounds on partial derivative complexity, and one of the first applications of these techniques outside the context of Sylvester-Gallai type problems.

Cite As Get BibTex

Nir Shalmon and Amir Shpilka. Partial Derivative Complexity of a Product of Linearly Independent Quadratics. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 152:1-152:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.152

Author Details

Nir Shalmon
  • Blavatnik School of Computer Science and AI, Tel Aviv University, Israel
Amir Shpilka
  • Blavatnik School of Computer Science and AI, Tel Aviv University, Israel

Funding

This research was funded by the European Union (ERC, EACTP, 101142020). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

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