The saturation number of \(\mathbf{c}\)-bounded stable monomial ideals and their powers
In this paper, the authors mainly investigate the behavior of the saturation number of \(\mathbf{c}\)-bounded strongly stable ideals and their powers. They obtain relevant results for such a class of monomial ideals. In general, let \(I\) be a monomial ideal of a polynomial ring over a field, \(S=K[x_1,\ldots,x_n]\), it is known that the saturation number of \(I^k\), \(\mathrm{sat}(I^k)\), is a quasilinear function for \(k\gg 0\). The authors prove that \(\mathrm{sat}(I^k)\) is a quasilinear funtion from the beginning for a particular class of monomials ideals. In detail, the authors firstly compute in a simple way the saturation number of strongly stable ideals. Moreover, they compute the socle and the saturation number of equigenerated \(\mathbf{c}\)-bounded strongly stable ideals. We recall that a \(\mathbf{c}\)-bounded ideal is an ideal of \(S\) generated by monomials \(\mathbf{x}^\mathbf{a}\), with \(\mathbf{a}=(a_1,\ldots, a_n)\) bounded by \(\mathbf{c}=(c_1,\ldots, c_n)\). These results lead to an explicit formula for \(\text{sat}(I)\) when \(I\) is an ideal of Veronese type, \textit{i.e.}, a particular type of \(\mathbf{c}\)-bounded strongly stable ideal. Finally, the authors prove that \(\mathrm{sat}(I^k)\) is a quasilinear function from the beginning for a Veronese-type ideal \(I\), and they also provide an explicit formula for such a quasilinear function. In this work interesting examples are given to illustrate the constructions and the necessity of the hypotheses.
\(\mathbf{c}\)-bounded strongly stable ideal
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