Abstract | We study the uniformity of Buchberger algorithms for computing Gröbner bases with respect to a natural number parameter k in the exponents of the input polynomials. The problems is motivated by positive results of T. Takahashi on special exponential parameter series of polynomial sets in singularity theory. For arbitrary input sets uniformity is in general impossible. By the way of contrast we show that the Buchberger algorithm is indeed uniform up to a fiite case distinction on the exponential parameter k for inputs consisting of monomials and binomials only. Under this hypothesis the case distinction is algorithmic and partitions the parameter range into Presburger sets. In each case the Buchberger algorithm is uniform and can be described explicitly and algorithmically. In the course of the algorithm the exponential parameter k enters also the coefficients as exponent. Thus the uniformity in k is established with respet to parametric exponents in both terms and coefficients. These results are obtained as a consequence of a much more general theorem concerning Buchberger algorithms for sets of monomials and binomials with arbitrary parametric coefficients and exponents, generalizing the construction of Gröbner systems. |