Hilbert series and non-associative Gröbner bases
A grading of finite type on a set \(X\) is a map \(d\) from \(X\) into the positive integers such that \(X_r=\{x\in X \mid d(x)=r\}\) is finite for all \(r\). The generating series of \(X\) is \(G_X=\sum_{r=1}^{\infty}\#X_r\cdot t^r\). Let \(K\) be a field and \(K\{X\}\) the absolutely free (non-associative) \(K\)-algebra with unit generated by \(X\). Any grading of finite type on \(X\) turns \(K\{X\}\) into a graded algebra whose Hilbert series can be computed as \(1+q(G_X)\), where \(q(t)={1\over 2}(1-\sqrt{1-4t})= \sum_{n=0}^{\infty}{1\over n+1}{2n\choose n}t^{n+1}\) (Proposition 1.1). Let \(J\) be a proper homogeneous ideal in \(K\{X\}\). Each minimal Gröbner basis \(\Gamma\) of \(J\) (relative to an admissible order) becomes a graded set of finite type if one sets \(d(g)\) to be the degree of the leading monomial of \(g\in\Gamma\). Then the Hilbert series of the graded algebra \(K\{X\}/J\) can be computed as \(1+q(G_X-G_\Gamma)\) (Proposition 3.1). As an application, the author shows that if \(\Gamma\) is a minimal Gröbner basis for the kernel \(J_a\) of the natural projection of \(K\{X\}\) onto the free associative \(K\)-algebra with unit generated by \(X\), then \(G_\Gamma=G_X^3(1-G_X)^{-2}\) (Proposition 4.1). He also gives an explicit construction for a minimal Gröbner basis of \(J_a\) (Proposition 4.2).
scientific article; zbMATH DE number 1556761