Degree bounds for Gröbner bases of modules (Q2066953)

Let \(R=K[x_1,\dots ,x_n]\) be the polynomial ring in \(n\) variables over the field \(K\). Let us fix a monomial ordering \(\prec\) on \(R\). Furthermore, let \(I\subset R\) be an ideal generated by a set of homogeneous polynomials of degree at most \(d\). Finally let \(D\) be the Krull dimension of \(R/I\). One of the main challenges in the theory of Gröbner bases, is to give an upper bound on degrees of polynomials in the reduced Gröbner basis of \(I\) with respect to \(\prec\) in terms of \(n,d\) and \(D\). \textit{T. W. Dubé} [SIAM J. Comput. 19, No. 4, 750--773 (1990; Zbl 0697.68051)] proved the upper bound \(2(d^2/2+d)^{2^{n-1}}\) and \textit{E. W. Mayr} and \textit{S. Ritscher} [J. Symb. Comput. 49, 78--94 (2013; Zbl 1258.13032)] improved it to \(2(1/2(d^{n-D}+d))^{2^{D-1}}\) for the maximum degree of the elements of the reduced Gröbner basis of \(I\). In the paper under review, by generalizing the ideas of Dubé and Mayr-Ritscher, the author shows the following upper bound on the Gröbner basis degrees of a graded submodule of a free module: Let \(F\) be a free \(R\)-module with basis elements \(\{\mathbf{e}_1,\dots, \mathbf{e}_m\}\) so that \(\deg(\mathbf{e}_j) \ge 0\) for all \(j\) and \(l = \max\{\deg(\mathbf{e}_1),\dots ,\deg(\mathbf{e}_m)\}\). Let \(M\) be a graded submodule of \(F\) generated by homogeneous elements with maximum degree \(d\ge \max\{1,l\}\) and \(\dim(F/M) = D\). Then the degree of the reduced Gröbner basis of \(M\) for any module monomial ordering on \(F\) is bounded by \(dmn-n+1\) if \(D=0\) and \(2(1/2((dm)^{n-D}m+d))^{2^{D-1}}\) otherwise.