scholarly journals Ideals Whose Hilbert Function and Hilbert Polynomial Agree at n = 1

1993 ◽  
Vol 157 (2) ◽  
pp. 534-547 ◽  
Author(s):  
J.D. Sally
Keyword(s):  
Hilbert Function ◽  
Hilbert Polynomial

scholarly journals Tight closure of powers of ideals and tight hilbert polynomials

2019 ◽  
Vol 169 (2) ◽  
pp. 335-355
Author(s):  
KRITI GOEL ◽  
J. K. VERMA ◽  
VIVEK MUKUNDAN
Keyword(s):  
Local Ring ◽  
Hilbert Function ◽  
Simplicial Complexes ◽  
Primary Ideal ◽  
Hilbert Polynomial ◽  
Prime Characteristic ◽  
Tight Closure ◽  
Powers Of Ideals ◽  
Hilbert Polynomials

AbstractLet (R, ) be an analytically unramified local ring of positive prime characteristic p. For an ideal I, let I* denote its tight closure. We introduce the tight Hilbert function $$H_I^*\left( n \right) = \Im \left( {R/\left( {{I^n}} \right)*} \right)$$ and the corresponding tight Hilbert polynomial $$P_I^*\left( n \right)$$, where I is an m-primary ideal. It is proved that F-rationality can be detected by the vanishing of the first coefficient of $$P_I^*\left( n \right)$$. We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes.

scholarly journals On Hilbert polynomial of certain determinantal ideals

1991 ◽  
Vol 14 (1) ◽  
pp. 155-162 ◽  
Author(s):  
Shrinivas G. Udpikar
Keyword(s):  
Polynomial Ring ◽  
Invariant Theory ◽  
Fundamental Theorem ◽  
Hilbert Function ◽  
Hilbert Polynomial ◽  
Determinantal Ideals ◽  
The Ideal

LetX=(Xij)be anm(1)bym(2)matrix whose entriesXij,1≤i≤m(1),1≤j≤m(2); are indeterminates over a fieldK. LetK[X]be the polynomial ring in thesem(1)m(2)variables overK. A part of the second fundamental theorem of Invariant Theory says that the idealI[p+1]inK[X], generated by(p+1)by(p+1)minors ofXis prime. More generally in [1], Abhyankar defines an idealI[p+a]inK[X], generated by different size minors ofXand not only proves its primeness but also calculates the Hilbert function as well as the Hilbert polynomial of this ideal. The said Hilbert polynomial is completely determined by certain integer valued functionsFD(m,p,a). In this paper we prove some important properties of these integer valued functions.

scholarly journals Hilbert polynomials of the algebras of $SL_ 2$-invariants

2018 ◽  
Vol 10 (2) ◽  
pp. 303-312
Author(s):  
N.B. Ilash
Keyword(s):  
Quadratic Forms ◽  
Hypergeometric Series ◽  
Hilbert Function ◽  
Dimensional Space ◽  
Hilbert Polynomial ◽  
Binary Forms ◽  
Hilbert Polynomials ◽  
Joint Invariants ◽  
Narayana Numbers ◽  
Classical Invariant

We consider one of the fundamental problems of classical invariant theory, the research of Hilbert polynomials for an algebra of invariants of Lie group $SL_2$. Form of the Hilbert polynomials gives us important information about the structure of the algebra. Besides, the coefficients and the degree of the Hilbert polynomial play an important role in algebraic geometry. It is well known that the Hilbert function of the algebra $SL_n$-invariants is quasi-polynomial. The Cayley-Sylvester formula for calculation of values of the Hilbert function for algebra of covariants of binary $d$-form $\mathcal{C}_{d}= \mathbb{C}[V_d\oplus \mathbb{C}^2]_{SL_2}$ (here $V_d$ is the $d+1$-dimensional space of binary forms of degree $d$) was obtained by Sylvester. Then it was generalized to the algebra of joint invariants for $n$ binary forms. But the Cayley-Sylvester formula is not expressed in terms of polynomials.In our article we consider the problem of computing the Hilbert polynomials for the algebras of joint invariants and joint covariants of $n$ linear forms and $n$ quadratic forms. We express the Hilbert polynomials $\mathcal{H} \mathcal{I}^{(n)}_1,i)=\dim(\mathcal{C}^{(n)}_1)_i, \mathcal{H}(\mathcal{C}^{(n)}_1,i)=\dim(\mathcal{C}^{(n)}_1)_i,$ $\mathcal{H}(\mathcal{I}^{(n)}_2,i)=\dim(\mathcal{I}^{(n)}_2)_i, \mathcal{H}(\mathcal{C}^{(n)}_2,i)=\dim(\mathcal{C}^{(n)}_2)_i$ of those algebras in terms of quasi-polynomial. We also present them in the form of Narayana numbers and generalized hypergeometric series.

scholarly journals Multigraded Castelnuovo–Mumford regularity via Klyachko filtrations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Rosa M. Miró-Roig ◽  
Martí Salat-Moltó
Keyword(s):  
Polynomial Ring ◽  
Toric Variety ◽  
Hilbert Function ◽  
Hilbert Polynomial ◽  
Lattice Polytopes ◽  
Sharp Bounds ◽  
Graded Modules ◽  
Cox Ring ◽  
Regularity Index

Abstract In this paper, we consider Z r \mathbb{Z}^{r} -graded modules on the Cl ⁡ ( X ) \operatorname{Cl}(X) -graded Cox ring C ⁢ [ x 1 , … , x r ] \mathbb{C}[x_{1},\dotsc,x_{r}] of a smooth complete toric variety 𝑋. Using the theory of Klyachko filtrations in the reflexive case, we construct a collection of lattice polytopes codifying the multigraded Hilbert function of the module. We apply this approach to reflexive Z s + r + 2 \mathbb{Z}^{s+r+2} -graded modules over any non-standard bigraded polynomial ring C ⁢ [ x 0 , … , x s , y 0 , … , y r ] \mathbb{C}[x_{0},\dotsc,x_{s},\allowbreak y_{0},\dotsc,y_{r}] . In this case, we give sharp bounds for the multigraded regularity index of their multigraded Hilbert function, and a method to compute their Hilbert polynomial.

scholarly journals Sharp upper bounds for the Betti numbers of a given Hilbert polynomial

2013 ◽  
Vol 7 (5) ◽  
pp. 1019-1064 ◽  
Author(s):  
Giulio Caviglia ◽  
Satoshi Murai
Keyword(s):  
Betti Numbers ◽  
Upper Bounds ◽  
Hilbert Polynomial

scholarly journals A trace formula for vector-valued modular forms

2015 ◽  
Vol 17 (06) ◽  
pp. 1550069
Author(s):  
P. Bantay
Keyword(s):  
Trace Formula ◽  
Modular Forms ◽  
Weight Distribution ◽  
Hilbert Polynomial ◽  
Free Generators ◽  
Vector Valued

We present a formula for vector-valued modular forms, expressing the value of the Hilbert-polynomial of the module of holomorphic forms evaluated at specific arguments in terms of traces of representation matrices, restricting the weight distribution of the free generators.

Sign in / Sign up

Export Citation Format

Share Document