Questions and conjectures on extremal Hilbert series

Fröberg, Ralf; Lundqvist, Samuel

doi:10.33044/revuma.v59n2a10

Cited by 9 publications

(2 citation statements)

References 23 publications

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“…We now give a conjecture in the case when the f i 's are generic, c.f. [9]. To prove the conjecture for some fixed (d 1 , .…”

Section: Introductionmentioning

confidence: 99%

Hilbert series of generic ideals in products of projective spaces

Fröberg¹

2021

Preprint

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If k[x 1 , . . . , x n ]/I = R = i≥0 R i , k a field, is a standard graded algebra, the Hilbert series of R is the formal power series i≥0 dim k R i t i . It is known already since Macaulay which power series are Hilbert series of graded algebras [12]. A much harder question is which series are Hilbert series if we fix the number of generators of I and their degrees, say for ideals I = (f 1 , . . . , f r ), deg f i = d i , i = 1, . . . , r. In some sense "most" ideals with fixed degrees of their generators have the same Hilbert series. There is a conjecture for the Hilbert series of those "generic" ideals, see below. In this paper we make a conjecture, and prove it in some cases, in the case of generic ideals of fixed degrees in the coordinate ring of P 1 × P 1 , which might be easier to prove.

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“…We now give a conjecture in the case when the f i 's are generic, c.f. [9]. To prove the conjecture for some fixed (d 1 , .…”

Section: Introductionmentioning

confidence: 99%

Hilbert series of generic ideals in products of projective spaces

Fröberg¹

2021

Preprint

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“…This question is inspired by the Fröberg conjecture [5] on the minimal Hilbert series of the quotient of a the polynomial ring with a homogeneous ideal. For a review of the Fröberg conjecture and related problems, see [6]. In this note we will focus of subalgebras generated in degree two, with minimal Hilbert function.…”

Section: Introductionmentioning

confidence: 99%

Subalgebras generated in degree two with minimal Hilbert function

Nicklasson

2019

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What can be said about the subalgebras of the polynomial ring, with minimal or maximal Hilbert function? This question was discussed in a recent paper by M. Boij and A. Conca. In this paper we study the subalgebras generated in degree two with minimal Hilbert function. The problem to determine the generators of these algebras transfers into a combinatorial problem on counting maximal north-east lattice paths inside a shifted Ferrers diagram. We conjecture that the subalgebras generated in degree two with minimal Hilbert function are generated by an initial Lex or RevLex segment.

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