Normal and Triangular Determinantal Representations of Multivariate Polynomials

In this paper we give a new and simple algorithm to put any multivariate polynomial into a normal determinant form in which each entry has the form , and in each column the same variable appears. We also apply the algorithm to obtain a triangular determinant representation, a reduced determinant representation, and a uniform determinant representation of any multivariable polynomial. The algorithm could be useful for obtaining representations of dimensions smaller than those available up to now to solve numerical problems.