Affine Projections of Polynomials - Kayal, Neeraj (Symposium on Theory of Computing (STOC), 2012)
An m-variate polynomial f is said to be an affine projection of some n-variate polynomial g if there exists an nm matrix A and an n-dimensional vector b such that f(x)=g(Ax+b). In other words, if f can be obtained by replacing each variable of g by an affine combination of the variables occurring in f, then it is said to be an affine projection of g. Some well known problems (such as the determinant versus permanent and matrix multiplication for example) are instances of this problem. Given f and g can we determine whether f is an affine projection of g? The intention of this paper is to understand the complexity of the corresponding computational problem: given polynomials f and g find A and b such that f=g(Ax+b), if such an (A,b) exists. We first show that this is an NP-hard problem. We then focus our attention on instances where g is a member of some fixed, well known family of polynomials so that the input consists only of the polynomial f(x) having m variables and degree d. We consider the situation where f(x) is given to us as a blackbox (i.e. for any point a in Fm we can query the blackbox and obtain f(a) in one step) and devise randomized algorithms with running time poly(mnd) in the following special cases. Firstly where g is the Permanent (respectively the Determinant) of an nxn matrix and A is of rank n2. Secondly where g is the sum of powers polynomial (respectively the sum of products polynomial), and A is a random matrix of the appropriate dimensions (also d should not be too small).