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The Complexity of Factors of Multivariate Polynomials
Citation key B-The-Complexity-Of-Factors-Of-Multivariate-Polynomials-1
Author Peter Bürgisser
Title of Book Proceedings 42nd FOCS
Pages 378-385
Year 2001
Address Las Vegas
Month October 14-17
Abstract The existence of string functions, which are not polynomial time computable, but whose graph is checkable in polynomial time, is a basic assumption in cryptography. We prove that in the framework of algebraic complexity, there are no such families of polynomial functions of p-bounded degree over fields of characteristic zero. The proof relies on a polynomial upper bound on the approximative complexity of a factor g of a polynomial f in terms of the (approximative) complexity of f and the degree of the factor g. This extends a result by Kaltofen (STOC 1986). The concept of approximative complexity allows to cope with the case that a factor has an exponential multiplicity, by using a perturbation argument. Our result extends to randomized (two-sided error) decision complexity.
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