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Ralf Fröberg

Publications and source records attributed to Ralf Fröberg.

At least 19 recordsLinked to original sources
arXiv2026

Symbolic powers of the ideal of$n$ general points in $P^{n-1}$

Problem L of Fröberg--Lundqvist--Oneto--Shapiro asks for the difference between the Hilbert series of ordinary and symbolic powers of the ideal of general points in projective space. We solve this completely for \(n\) general points of \(\PP^{n-1}\). Besides a closed formula for \[ \HS(S/I^m)-\HS(S/I^{(m)}), \] we determine all minimal monomial generators of \(I^{(m)}\), and describe the symbolic Rees algebra. We also show that containment \(I^{(m)}\subseteq I^r\) is detected solely by initial degrees. This gives the exact containment threshold, the Waldschmidt constant \(\walpha\), the resurgence \(\Res\), and the asymptotic resurgence \(\aRes\): \[ \walpha(I)=\frac{n}{n-1}, \qquad \Res(I)=\aRes(I)=\frac{2(n-1)}{n}. \] We also take the first step beyond \(n\) points: for \(n+1\) general points of \(\PP^{n-1}\) --- again a rigid, non-monomial configuration --- we identify the defining quadrics, resolve the case \(n=3\) completely (a complete intersection, with \(J^{(m)}=J^m\) for all \(m\) and resurgence \(1\)), and propose an exact Waldschmidt-constant formula \(\walpha=\frac{n+1}{n-1}\) for all \(n\), verified computationally in every case we could check.

math.AC↗
arXiv2026

Generic forms

We study forms $I=(f_1,\ldots,f_r)$, $°f_i=d_i$, in $F$ which is the free associative algebra $k\langle x_1,\ldots,x_n\rangle$ or the polynomial ring $k[x_1,\ldots,x_n]$, where $k$ is a field and $°x_i=1$ for all $i$. We say that $I$ has type $t=(n;d_1,\ldots,d_r)$ and also that $F/I$ is a $t$-presentation. For each prime field $k_0$ and type $t=(n;d_1,\ldots,d_r)$, there is a series which is minimal among all Hilbert series for $t$-presentations over fields with prime field $k_0$ and such a $t$-presentation is called generic if its Hilbert series coincides with the minimal one. When the field is the real or complex numbers, we show that a $t$-presentation is generic if and only if it belongs to a non-empty countable intersection $C$ of Zariski open subsets of the affine space, defined by the coefficients in the relations, such that all points in $C$ have the same Hilbert series. In the commutative case there is a conjecture on what this minimal series is, and we give a conjecture for the generic series in the non-commutative quadratic case (building on work by Anick). We prove that if $A=k\langle x_1,\ldots,x_n\rangle/(f_1,\ldots,f_r)$ is a generic quadratic presentation, then $\{ x_if_j\}$ either is linearly independent or generate $A_3$. This complements a similar theorem by Hochster-Laksov in the commutative case. Finally we show, a bit to our surprise, that the Koszul dual of a generic presentation is not generic in general. But if the relations have algebraically independent coefficients over the prime field, we prove that the Koszul dual is generic. Hereby, we give a counterexample of \cite[Proposition 4.2]{P-P}, which states a criterion for a generic non-commutative quadratic presentation to be Koszul. We formulate and prove a correct version of the proposition.

math.AC↗
arXiv2026

Betti numbers of split graphs

A split graph is a graph where the vertices are a disjoint union of a complete part $C=\{x_i,\ldots,x_n\}$ and a stable part $S=\{y_1,\ldots,y_m\}$. We will determine the Betti numbers of the edge ring of all split graphs, in particular show that the only nonzero Betti numbers are $β_{0,0}$ and $β_{i,i+1}$, $i>0$. The Betti numbers only depend on the multiset of the number of neighbors in $S$ the $x_i$'s have. Singh and Verma have earlier determined the Betti numbers for complete split graphs (where all $y_i$ are neighbors to all $x_j$), and for "nearly complete" split graphs (where all $y_i$ are neighbors to all $x_j$, except that $y_i$ is not a neighbor to $x_i$ for $i=1,\ldots,\min\{m,n\}$). We also determine which split graphs that have Cohen-Macaulay edge ring.

math.AC↗
arXiv2026

Betti numbers of skeletons of thick trees

The starting point is the class of the following simplicial complexes $Δ$ with 2-linear resolutions. The facets of $Δ$ are $F_1,\ldots,F_n$, and we demand that for each $i$ $F_i\cap (F_1\cup \cdots\cup F_{i-1}\cup F_{i+1}\cdots\cup F_n)$ be a point. We will determine the Betti numbers, and thus the projective dimension, the depth, and the regularity of the Stanley-Reisner rings of all skeletons of such complexes. It follows that we know when these complexes are Cohen-Macaulay. Also, there are two ways to determine the Hilbert series of $Δ$, giving sequences of identities for binomial coefficients.

math.AC↗
arXiv2026

Some new Betti numbers of ideals generated by n+1 generic forms in n variables

Very little is known on the Hilbert series of graded algebras $\mathbb C[x_1,\ldots,x_n]/(g_1,\ldots,g_r)$, $r>n$, $g_i$ generic form of degree $e_i$, in general. One instance when the series is known, is for $n+1$ forms in $n$ variables, \cite{St}. Of course even less is known about Betti numbers. There are some general results on the Betti table by Pardue and Richert in \cite{Pa-Ri,Pa-Ri1}, and by Diem in \cite{Di}. Then there are results on Betti numbers in the case $n+1$ relations in $n$ variables, described below, by Migliore and Mirò-Roig in \cite{Mi-Mi}, and more partial results in the general case by the same authors in \cite{Mi-Mi1}. In this paper we consider the same case as in \cite{Mi-Mi}, $n+1$ forms in $n$ variables. Our results can be described as follows. We can determine all graded Betti numbers of $\mathbb C[x_1,\ldots,x_n]/(g_1,\ldots,g_{n+1})$, $g_i$ generic, at least if $\sum_{i=1}^{n+1}°(g_i)-n$ is even, often in more cases. Thus, given {\em any} set $\{ e_1,\ldots,e_n\}$, $e_i\ge2$ for all $i$, such that $°(g_i)=e_i$, $i=1,\ldots,n$, we get many numbers $D_j$, so that we can determine all graded Betti numbers of $\mathbb C[x_1,\ldots,x_n]/(g_1,\ldots,g_{n+1})$, $°(g_i)=e_i$, $1\le i\le n$, $°(g_{n+1})=D_j$. The main ingredients of the proof is a theorem by Pardue and Richert, \cite{Pa-Ri,Pa-Ri1}, and later by Diem,\cite{Di}, and a new short proof of a theorem on Hilbert series of artinian complete intersections by Reid, Roberts, and Roitman, \cite{R-R-R}. We also give examples of algebras with many so called "ghost terms" in the minimal resolution.

math.AC↗
arXiv2022

The graded Betti numbers of truncation of ideals in polynomial rings

Let $R=\mathbb{K}[x_1,\dots,x_n]$, a graded algebra $S=R/I$ satisfies $N_{k,p}$ if $I$ is generated in degree $k$, and the graded minimal resolution is linear the first $p$ steps, and the $k$-index of $S$ is the largest $p$ such that $S$ satisfies $N_{k,p}$. Eisenbud and Goto have shown that for any graded ring $R/I$, then $R/I_{\geq k}$, where $I_{\geq k}=I\cap M^k$ and $M=(x_1,\dots,x_n)$, has a $k$-linear resolution (satisfies $N_{k,p}$ for all $p$) if $k\gg0$. For a squarefree monomial ideal $I$, we are here interested in the ideal $I_k$ which is the squarefree part of $I_{\geq k}$. The ideal $I$ is, via Stanley-Reisner correspondence, associated to a simplicial complex $Δ_I$. In this case, all Betti numbers of $R/I_k$ for $k>\min\{\text{deg}(u)\mid u\in I\}$, which of course is a much finer invariant than the index, can be determined from the Betti diagram of $R/I$ and the $f$-vector of $Δ_I$. We compare our results with the corresponding statements for $I_{\ge k}$. (Here $I$ is an arbitrary graded ideal.) In this case we show that the Betti numbers of $R/I_{\ge k}$ can be determined from the Betti numbers of $R/I$ and the Hilbert series of $R/I_{\ge k}$.

math.AC↗
arXiv2022

Solution to a conjecture on edge rings with 2-linear resolutions

For a graph $G=(V,E)$ the edge ring $k[G]$ is $k[x_1,\ldots,x_n]/I(G)$, where $n=|V|$ and $I(G)$ is generated by $\{ x_ix_j;\{ i,j\}\in E\}$. The conjecture we treat is the following. If $k[G]$ has a 2-linear resolution, then the projective dimension of $K[G]$, pd$(k[G])$, equals the maximal degree of a vertex in $G$. As far as we know, this conjecture is first mentioned in a paper by Gitler and Valencia, and there it is called the Eliahou-Villarreal conjecture. The conjecture is treated in a recent paper by Ahmed, Mafi, and Namiq. That there are counterexamples was noted already by Moradi and Kiani. By interpreting $k[G]$ as a Stanley-Reisner ring, we are able to characterize those graphs for which the conjecture holds.

math.AC↗
arXiv2022

Betti numbers of fat forests and their Alexander dual

Let $k$ be a field and $R=k[x_1,\ldots,x_n]/I=S/I$ a graded ring. Then $R$ has a $t$-linear resolution if $I$ is generated by homogeneous elements of degree $t$, and all higher syzygies are linear. Thus $R$ has a $t$-linear resolution if ${\rm Tor}^S_{i,j}(S/I,k)=0$ if $j\ne i+t-1$. For a simplicial complex $Δ$ on $[{\bf n}]=\{1,\ldots,n\}$ and a field $k$, the Stanley-Reisner ring $k[Δ]$ is $k[x_1,\ldots,x_n]/I$, where $I$ is generated by those squarefree monomials $x_{i_1}\cdots x_{i_k}$ for which $\{ i_1,\ldots,i_k\}$ does not belong to $Δ$. In \cite{Fr} the Stanley-Reisner rings with 2-linear resolution are determined. Their associated complexes has had different names in the literature. We call them fat forests here. In this article we determine the Betti numbers of fat forests. We also consider Betti numbers of Alexander duals of fat forests.

math.AC↗
arXiv2017

Extremal Hilbert series

Given an ideal of forms in an algebra (polynomial ring, tensor algebra, exterior algebra, Lie algebra, bigraded polynomial ring), we consider the Hilbert series of the factor ring. We concentrate on the minimal Hilbert series, which is achieved when the forms are generic. In the polynomial ring we also consider the opposite case of maximal series. This is mainly a survey article, but we give a lot of problems and conjectures. The only novel results concern the maximal series in the polynomial ring.

math.AC↗
arXiv2016

Powers of generic ideals and the weak Lefschetz property for powers of some monomial complete intersections

Given an ideal $I=(f_1,\ldots,f_r)$ in $\mathbb C[x_1,\ldots,x_n]$ generated by forms of degree $d$, and an integer $k>1$, how large can the ideal $I^k$ be, i.e., how small can the Hilbert function of $\mathbb C[x_1,\ldots,x_n]/I^k$ be? If $r\le n$ the smallest Hilbert function is achieved by any complete intersection, but for $r>n$, the question is in general very hard to answer. We study the problem for $r=n+1$, where the result is known for $k=1$. We also study a closely related problem, the Weak Lefschetz property, for $S/I^k$, where $I$ is the ideal generated by the $d$'th powers of the variables.

math.AC↗
arXiv2013

On free resolutions of some semigroup rings

For some numerical semigroup rings of small embedding dimension, namely those of embedding dimension 3, and symmetric or pseudosymmetric of embedding dimension 4, presentations has been determined in the literature. We extend these results by giving the whole graded minimal free resolutions explicitly. Then we use these resolutions to determine some invariants of the semigroups and certain interesting relations among them. Finally, we determine semigroups of small embedding dimensions which have strongly indispensable resolutions.

math.AC↗
arXiv2013

Vandermonde varieties and relations among Schur polynomials

Motivated by the famous Skolem-Mahler-Lech theorem we initiate in this paper the study of a natural class of determinantal varieties which we call {\em Vandermonde varieties}. They are closely related to the varieties consisting of all linear recurrence relations of a given order possessing a non-trivial solution vanishing at a given set of integers. In the regular case, i.e., when the dimension of a Vandermonde variety is the expected one, we present its free resolution, obtain its degree and the Hilbert series. Some interesting relations among Schur polynomials are derived. Many open problems and conjectures are posed.

math.AG↗
arXiv2011

On the Waring problem for polynomial rings

In this note we discuss an analog of the classical Waring problem for C[x_0, x_1,...,x_n]. Namely, we show that a general homogeneous polynomial p \in C[x_0,x_1,...,x_n] of degree divisible by k\ge 2 can be represented as a sum of at most k^n k-th powers of homogeneous polynomials in C[x_0, x_1,...,x_n]. Noticeably, k^n coincides with the number obtained by naive dimension count.

math.AG↗
arXiv2011

On differential operators of numerical semigroup rings

If $S= $ is a numerical semigroup, we call the ring $\C[S]=\C[t^{d_1},...,t^{d_ν}]$ the semigroup ring of $S$. We study the ring of differential operators on $\C[S]$, and its associated graded in the filtration induced by the order of the differential operators. We find that these are easy to describe in case $S$ is a so called Arf semigroup. If $I$ is an ideal in $\C[S]$ that is generated by monomials, we also give some results on $\der(I,I)$ (the set of derivations which map $I$ into $I$).

math.AC↗