Evaluation Properties of Symmetric Polynomials

By the fundamental theorem of symmetric polynomials, if P Q[X1, . . . , Xn] is symmetric, then it can be written P = Q(1, . . . , n), where 1, . . . , n are the elementary symmetric polynomials in n variables, and Q is in Q[S1, . . . , Sn]. We investigate the complexity properties of this construction in the straight-line program model, showing that the complexity of evaluation of Q depends only on n and on the complexity of evaluation of P. Similar results are given for the decomposition of a general polynomial in a basis of Q[X1, . . . , Xn] seen as a module over the ring of symmetric polynomials, as well as for the computation of the Reynolds operator.

• Type: Journal
• Year: 2006
• Where: IJAC
• Authors: Pierrick Gaudry, Éric Schost, Nicolas M. Thiéry
• Keywords: Elementary Symmetric Polynomials, Fundamental Theorem, IJAC 2006, Symmetric Polynomials