Poincaré series of monomial rings with minimal Taylor resolution (Q2904274)

Let \(A=k[x_1,\dots,x_n]\) be a polynomial ring over a field \(K\) and \(I\) be an ideal of \(A.\) The Poincaré series of \(R=A/I\) is the power seriesNEWLINENEWLINENEWLINE\[NEWLINEP_k^R(z)=\sum_{i\geq 0}\dim_k(\text{Tor}_i^R(k,k))z^i\in \mathbb Z[[z]].NEWLINE\]NEWLINE It is the generating function of the sequence of Betti numbers of a minimal free resolution of \(k\) over \(R\). A question that this becomes a rational function was asked by Serre. An affirmative answer was presented by \textit{J. Backelin} [C. R. Acad. Sci. Paris Sér. I Math. 295, 607--610 (1982; Zbl 0518.13016)] when \(I\) is a monomial ideal in \(A\).NEWLINENEWLINEIn this paper the author compute the Poincaré series for a new class of some monomial ideals with minimal Taylor resolution where the ideal is generated by quadratic monomials. The author use the polarization of monomial rings introduced by \textit{R. Fröberg} [Math. Scand. 51, 22--34 (1982; Zbl 0498.13012)] to prove the result.