On the Complexity of Symmetric Polynomials

Authors Markus Bläser, Gorav Jindal

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Author Details

Markus Bläser
  • Department of Computer Science, Saarland University, Saarland Informatics Campus, Saarbrücken, Germany
Gorav Jindal
  • Department of Computer Science, Aalto University, Espoo, Finland

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Markus Bläser and Gorav Jindal. On the Complexity of Symmetric Polynomials. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 47:1-47:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


The fundamental theorem of symmetric polynomials states that for a symmetric polynomial f_{Sym} in C[x_1,x_2,...,x_n], there exists a unique "witness" f in C[y_1,y_2,...,y_n] such that f_{Sym}=f(e_1,e_2,...,e_n), where the e_i's are the elementary symmetric polynomials. In this paper, we study the arithmetic complexity L(f) of the witness f as a function of the arithmetic complexity L(f_{Sym}) of f_{Sym}. We show that the arithmetic complexity L(f) of f is bounded by poly(L(f_{Sym}),deg(f),n). To the best of our knowledge, prior to this work only exponential upper bounds were known for L(f). The main ingredient in our result is an algebraic analogue of Newton's iteration on power series. As a corollary of this result, we show that if VP != VNP then there exist symmetric polynomial families which have super-polynomial arithmetic complexity. Furthermore, we study the complexity of testing whether a function is symmetric. For polynomials, this question is equivalent to arithmetic circuit identity testing. In contrast to this, we show that it is hard for Boolean functions.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Symmetric Polynomials
  • Arithmetic Circuits
  • Arithmetic Complexity
  • Power Series
  • Elementary Symmetric Polynomials
  • Newton's Iteration


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