Comparing complexities of problems of determining of Grebner’s basis of ideal and solving this ideal.
A new method of investigation of ideals in the rings of polynomials was proposed by B. Buchberger in 1965. He proposed to use special basis in such rings named “Gröbner Basis” in the honor of his teacher. The proposed approach has allowed to prove the algorithmic reducibility of the task of finding solutions of a system of algebraic equations in many variables to the problem of solving algebraic equation of one variable. However, the practical use of the method needs enormous computational cost. The first estimate of the computational complexity of this method was obtained by G. Hermann in 1925, when the concept of Gröbner basis was not yet known. Was obtained the required upper estimate in the form of dual exponent of the number of variables and the maximum degree of incoming task description polynomials of degree of decomposition element ideal for an arbitrary basis. As it turned out later by T.W. Dube this estimate cannot be improved. In 1996 K. Kühnle and E. W. Mayr proved, that for Boolean ideals the upper bound of memory capacity is exponential on input size. For zero-dimensional ideals A. Hashemi and D. Lazard proved the exponential complexity of finding Gröbner basis. Here we prove the lower bound of computation of such bases, by giving an example of an ideal having Gröbner basis of exponential size on input.
Proceedings of the Institute for System Programming, vol. 22, 2012, pp. 463-474.
ISSN 2220-6426 (Online), ISSN 2079-8156 (Print).