Affine projections of polynomials

Abstract

An m-variate polynomial f is said to be an affine projection of some n-variate polynomial g if there exists an n × m 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. Given f and g can we determine whether f is an affine projection of g? Some well known problems (such as VP versus VNP and matrix multiplication for example) are instances of this problem.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 ∈ F m 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:(1) when f = Permn(Ax + b) and A satisfies rank(A) = n 2 .Here Permn is the permanent polynomial.(2) when f = Detn(Ax + b) and A satisfies rank(A) = n 2 .Here Detn is the determinant polynomial.(3) when f = Pow n,d (Ax + b) and A is a random n × m matrix with d = n Ω(1) . Here Pow n,d is the power-symmetric polynomial of degree d.(4) when f = SPS n,d (Ax + b) and A is a random (nd) × m matrix with n constant. Here SPS n,d is the sum-ofproducts polynomials of degree d with n terms.

Approaching the Chasm at Depth Four

“…Kayal [Kay12a] showed that ∂ =1 f ≤1 yields a lie algebra that can help efficiently determine if f is equivalent (via an affine change of variables) to the permanent (or determinant). For = ∞, note that ∂ =k f ≤ is precisely the ideal generated by the k-th order derivatives of f .…”

Reconstruction Algorithms for Sums of Affine Powers

“…The algorithmic study of Model 1.2 is usually attributed to Sylvester. We refer to [15] for the historical background and to section 1.3 of that book for a description of the algorithm (see also Kleppe [17] and Proposition 46 of Kayal [18]). Most of the subsequent work was devoted to the multivariate generalization 1 of Model 1.2, with much of the 20th century work focused on the determination of the Waring rank of generic polynomials [1,7,15].…”

Testing Equivalence of Polynomials under Shifts

“…Moreover, he provides randomized algorithms for the affine-equivalence problem when one of the polynomials is the Permanent or the Determinant and the affine transformations x → Ax + b are of a special form (in the case of Determinant and Permanent, the matrix A must be invertible). Kayal provides randomized algorithms for some other classes of homogeneous polynomials, and for more details we refer the reader to the paper [Kay12]. Our work is different from Kayal's work since in our setting we are only interested in shift-equivalences, and in this feature we are less general than Kayal's work, but we also consider larger classes of polynomials, in which case we are more general than Kayal's work.…”

“…The study of equivalences of general polynomials under affine transformations, which we refer to as affine-equivalence, was started by Kayal in [Kay12] (note that this generalizes the problem studied in [GK93]). We say that f and g are affine-equivalent if there exists a matrix A and a shift b such that f (x) = g(Ax + b).…”

“…This approach uses a lemma due to Carlini [Car06] (see also [Kay12,Lemma 17]) and an argument implicit in Kayal's work [Kay12, Section 7.3]. We now discuss and compare this alternative approach to ours.…”