## Abstract

The fundamental theorem of symmetric polynomials states that for a symmetric polynomial f_{Sym} ∈ C[x_{1}, x_{2}, . . ., x_{n}], there exists a unique “witness” f ∈ C[y1, y2, . . ., 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 6= 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.

Original language | English |
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Title of host publication | 10th Innovations in Theoretical Computer Science, ITCS 2019 |

Editors | Avrim Blum |

Publisher | Schloss Dagstuhl-Leibniz-Zentrum für Informatik |

Pages | 1-14 |

ISBN (Electronic) | 9783959770958 |

DOIs | |

Publication status | Published - 1 Jan 2019 |

MoE publication type | A4 Article in a conference publication |

Event | Innovations in Theoretical Computer Science Conference - San Diego, United States Duration: 10 Jan 2019 → 12 Jan 2019 Conference number: 10 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Volume | 124 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | Innovations in Theoretical Computer Science Conference |
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Abbreviated title | ITCS |

Country/Territory | United States |

City | San Diego |

Period | 10/01/2019 → 12/01/2019 |

## Keywords

- Arithmetic circuits
- Arithmetic complexity
- Elementary symmetric polynomials
- Newton’s iteration
- Power series
- Symmetric polynomials