On the Hilbert series of ideals generated by generic forms

ABSTRACT

There is a longstanding conjecture by Fröberg about the Hilbert series of the ring R∕I, where R is a polynomial ring, and I an ideal generated by generic forms. We prove this conjecture true in the case when I is generated by a large number of forms, all of the same degree. We also conjecture that an ideal generated by m’th powers of generic forms of degree d≥2 gives the same Hilbert series as an ideal generated by generic forms of degree md. We verify this in several cases. This also gives a proof of the first conjecture in some new cases.

KEYWORDS:

• Graded algebras
• generic forms
• Hilbert series

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13A02
• 13F20

Acknowledgments

I would like to thank Ralf Fröberg for introducing me to this subject, and for help with the Macaulay2 computations. I also thank Christian Gottlieb for helpful comments on my work.