# The Complexity of Factors of Multivariate Polynomials

## TL;DR

This paper proves that in algebraic complexity over characteristic zero fields, there are no polynomial families with checkable but not polynomial-time computable functions, extending previous results and using approximative complexity methods.

## Contribution

It establishes a polynomial upper bound on the approximative complexity of polynomial factors, extending Kaltofen's earlier work and including randomized decision complexity.

## Key findings

- No families of string functions with checkable graphs exist in this framework.
- Provides a polynomial bound on the approximative complexity of polynomial factors.
- Extends results to randomized decision complexity.

## 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.