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. The concept of approximative complexity allows us 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Change history
06 October 2020
Vladimir Lysikov kindly pointed out an error in the proof of Theorem��5.7. We provide here a corrected statement and its proof.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bürgisser, P. The Complexity of Factors of Multivariate Polynomials. Found Comput Math 4, 369–396 (2004). https://doi.org/10.1007/s10208-002-0059-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10208-002-0059-5