Ideals generated by quadrics exhibiting double exponential degrees
Let \(I=(h_1,\dots,h_s)\) be an ideal in the polynomial ring \(R=K[x_1,\dots,x_n]\) over a field. A measure of the computational cost of algorithms solving the ideal membership problem is obtained by bounding the degrees of coefficients \(g_i\) which espress elements \(h\in I\) as \(h=\sum g_ih_i\). Actually we have bounds which are doubly exponential in the number \(n\) of variables; on the other hand, \textit{E. W. Mayr} and \textit{A. R. Meyer} [Adv. Math. 46, 305-329 (1982; Zbl 0506.03007] constructed examples of elements in some polynomial rings, requiring coefficients whose degrees have a doubly exponential growth; it turns out that known bounds can be possibly refined, but not substantially improved. The examples of Mayr and Meyer (loc. cit.) are generated in degree \(5\). In the present paper, the author modifies these examples, providing ideals generated by quadrics for which the doubly exponential growth occurs.
scientific article; zbMATH DE number 1139776