Fröberg conjecture explained

In algebraic geometry, the Fröberg conjecture is a conjecture about the possible Hilbert functions of a set of forms. It is named after Ralf Fröberg, who introduced it in . The Fröberg–Iarrobino conjecture is a generalization introduced by .

Statement of Conjecture

Given generic homogeneous polynomials

g _{1,g} _{2,\ldots,g} _{k\in} C [x _{1,x} _{2,\ldots,x} _{n]} of degrees a _{1,a} _{2,\ldots,} a _{k} resp. Then the Hilbert Series of C [x _{1,x} _{2,\ldots,x} _{n]/\langle} g _{1,g} _{2,\ldots,} g _{k\rangle} is {( 1+ t + t ^{2+\ldots)} ^{n}{(1-t} ^{a _{1}} )( 1- t ^{a _{2}} ) … ( 1- t ^{a _{k}} )} truncated at its first negative term