The Complexity of Factors of Multivariate Polynomials. Bürgisser, P. *Foundations of Computational Mathematics*, 4(4):369-396, 2004. 2 Paper abstract bibtex 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.

@ARTICLE{B-The-Complexity-Of-Factors-Of-Multivariate-Polynomials,
JOURNAL={Foundations of Computational Mathematics},
TITLE={The Complexity of Factors of Multivariate Polynomials},
VOLUME={4},
AUTHOR={Peter Bürgisser},
YEAR={2004},
URL={http://www3.math.tu-berlin.de/algebra/work/comfa.pdf},
PAGES={369-396},
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.},
URL2={http://link.springer.com/article/10.1007/s10208-002-0059-5},
NUMBER={4}
}

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