On the VNP-Hardness of Some Monomial Symmetric Polynomials

Authors Radu Curticapean , Nutan Limaye , Srikanth Srinivasan

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Author Details

Radu Curticapean
  • IT University of Copenhagen, Denmark
  • Basic Algorithms Research Copenhagen, Denmark
Nutan Limaye
  • IT University of Copenhagen, Denmark
  • Basic Algorithms Research Copenhagen, Denmark
Srikanth Srinivasan
  • Department of Computer Science, Aarhus University, Denmark
  • On leave from Department of Mathematics, IIT Bombay, India

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Radu Curticapean, Nutan Limaye, and Srikanth Srinivasan. On the VNP-Hardness of Some Monomial Symmetric Polynomials. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 16:1-16:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


A polynomial P ∈ 𝔽[x_1,…,x_n] is said to be symmetric if it is invariant under any permutation of its input variables. The study of symmetric polynomials is a classical topic in mathematics, specifically in algebraic combinatorics and representation theory. More recently, they have been studied in several works in computer science, especially in algebraic complexity theory. In this paper, we prove the computational hardness of one of the most basic kinds of symmetric polynomials: the monomial symmetric polynomials, which are obtained by summing all distinct permutations of a single monomial. This family of symmetric functions is a natural basis for the space of symmetric polynomials (over any field), and generalizes many well-studied families such as the elementary symmetric polynomials and the power-sum symmetric polynomials. We show that certain families of monomial symmetric polynomials are VNP-complete with respect to oracle reductions. This stands in stark contrast to the case of elementary and power symmetric polynomials, both of which have constant-depth circuits of polynomial size.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Computing methodologies → Representation of polynomials
  • algebraic complexity
  • symmetric polynomial
  • permanent
  • Sidon set


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