Fröberg conjecture

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 [sv], who introduced it in Fröberg (1985, page 120). The Fröberg–Iarrobino conjecture is a generalization introduced by Anthony Iarrobino (1997).

Statement of Conjecture

Given generic homogeneous polynomials ${\displaystyle g_{1},g_{2},\ldots ,g_{k}\in \mathbb {C} [x_{1},x_{2},\ldots ,x_{n}]}$ of degrees ${\displaystyle a_{1},a_{2},\ldots ,a_{k}}$, respectively. Then the Hilbert series of ${\displaystyle \mathbb {C} [x_{1},x_{2},\ldots ,x_{n}]/\langle g_{1},g_{2},\ldots ,g_{k}\rangle }$ is

${\displaystyle {(1+t+t^{2}+\cdots )^{n}}{(1-t^{a_{1}})(1-t^{a_{2}})\cdots (1-t^{a_{k}})},}$

truncated at its first negative term.

References

• Fröberg, Ralf (1985), "An inequality for Hilbert series of graded algebras", Mathematica Scandinavica, 56 (2): 117–144, doi:10.7146/math.scand.a-12092, ISSN 0025-5521, MR 0813632
• Iarrobino, Anthony (1997), "Inverse system of a symbolic power. III. Thin algebras and fat points", Compositio Mathematica, 108 (3): 319–356, doi:10.1023/A:1000155612073, ISSN 0010-437X, MR 1473851