Lower bounds for polar and tensor ranks

Abstract

If A is a symmetric complex valued n-linear function on then the tensor rank of A. denoted by r 1(A), is defined to be the smallest nonnegative integer q such that A is expressible as a linear combination of q decomposables. The polar rank of A, denoted by r 1(A) is defined to be the smallest non-negative integer q such that A is expressible as a linear combination of q powers — where by a power we mean a symmetric n-linear function of the form fn where . We present several lower bounds for r 1(A) and rp (A) that involve only the coefficients of A. Within the context of the polynomial algebra our results provide estimates for the minimum number of linearly factorable degree n homogeneous polynomials necessary to represent a given homogeneous degree n polynomial, and for the minimum number of homogeneous nth powers necessary to represent a given degree n homogeneous polynomial.