Title: The Complexity of Factors of Multivariate Polynomials,
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 polynomially 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.