On the Hilbert depth of the Hilbert function of a finitely generated graded module

Abstract Let K be a field, A a standard graded K -algebra and M a finitely generated graded A -module. Inspired by our previous works, see [2] and [3], we study the invariant called Hilbert depth of h M , that is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mrow> <m:mo>hdepth</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>h</m:mi> </m:mrow> <m:mi>M</m:mi> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mo>max</m:mo> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mi>d</m:mi> <m:mo>:</m:mo> <m:munder> <m:mo>∑</m:mo> <m:mrow> <m:mi>j</m:mi> <m:mo>≤</m:mo> <m:mi>k</m:mi> </m:mrow> </m:munder> <m:mrow> <m:msup> <m:mrow> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo>-</m:mo> <m:mi>j</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mtable> <m:mtr> <m:mtd> <m:mrow> <m:mi>d</m:mi> <m:mo>-</m:mo> <m:mi>j</m:mi> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mrow> <m:mi>k</m:mi> <m:mo>-</m:mo> <m:mi>j</m:mi> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>h</m:mi> </m:mrow> <m:mi>M</m:mi> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mi>j</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>≥</m:mo> <m:mn>0</m:mn> <m:mi> </m:mi> <m:mi> </m:mi> <m:mo>for all </m:mo> <m:mi>k</m:mi> <m:mo>≤</m:mo> <m:mi>d</m:mi> </m:mrow> </m:mrow> <m:mo>}</m:mo> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> \mathrm{hdepth} \left( {{h_M}} \right) = \max \left\{ {d:\sum\limits_{j \le k} {{{\left( { - 1} \right)}^{k - j}}\left( {\matrix{ {d - j} \cr {k - j} \cr } } \right){h_M}\left( j \right) \ge 0\,\, \mathrm{for}\, \mathrm{all}\, k \le d}} \right\}, where h M (−) is the Hilbert function of M , and we prove basic results regard it. Using the theory of hypergeometric functions, we prove that hdepth( h S ) = n , where S = K [ x 1 , . . . , x n ]. We show that hdepth( h S/J ) = n , if J = ( f 1 , . . . , f d ) ⊂ S is a complete intersection monomial ideal with deg(f i ) ≥ 2 for all 1 ≤ i ≤ d . Also, we show that hdepth( h M̅ ) ≥ hdepth( h M ) for any finitely generated graded S -module M , where M̅ = M ⊗ S S [x n +1 ].