In this paper, we present a new and simple algorithm for writing any multivariate polynomial in a new representation defined as a normal determinantal representation in which each entry has the form a_ix_i + b_i and the same variable appears in each column. We apply this algorithm to obtain the triangular, reduced, and uniform determinantal representations of any multivariate polynomial. The proposed algorithm could be useful also for obtaining smaller representations than those currently available for solving numerical problems.

determinantal representation, normal determinantal representation, triangular determinantal representation, uniform determinantal representation, multivariate polynomials, algorithm.