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 $g_1,g_2,\dots ,g_k\in \mathbb{ℂ}[x_1,x_2,\dots ,x_n]$ of degrees $a_1,a_2,\dots ,a_k$, respectively. Then the Hilbert series of $\mathbb{ℂ}[x_1,x_2,\dots ,x_n]/⟨g_1,g_2,\dots ,g_k⟩$ is

$(1+t+t^2+\cdots {)}^n(1-t^{a_1})(1-t^{a_2})\cdots (1-t^{a_k}),$

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 
• 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