Ideals Generated By Polynomials With Parametric Exponents

For a class of ideals generated by polynomials with parametric exponents, we first present the consistency condition that provides a sufficient condition for the construction of uniform Grobner bases. Then by analyzing the first syzygy module of a set of monomials, we propose the weak consistency condition which is also sufficient for constructing uniform Grobner bases. Moreover, we explain why the weak consistency condition is indeed weak.Finally, we apply the consistency condition to deal with free resolutions of monomial ideals. Application of uniform free resolutions shows what family of polynomials with parametric exponents is good.