Polynomial expressions for non-binomial structures
Let \(R=K[x_1,\ldots ,x_n]\) be a polynomial ring over a field \(K\) and \(f,f_1,\ldots ,f_k\in R\). Let us fix a term ordering on \(R\). The classical division algorithm claims that one is able to write \(f\) as an expression like \(g_1f_1+\cdots +g_kf_k+h\) where either \(h=0\) or no term in \(h\) is divisible by the leading terms of the \(f_i\)'s. In this paper, the author presents a framework for performing the division. For this purpose, he describes a process for dividing a polynomial by a single polynomial and then he uses this process for dividing a polynomial by a list of polynomials. In addition, application of the new algorithm on moderately large polynomials is discussed. The paper is concluded by a complexity analysis of some intermediate procedures.
scientific article; zbMATH DE number 7030854