V-numbers of powers of cover ideals of unimodular hypergraphs

Abstract: Let $H$ be a unimodular hypergraph with cover ideal $J(H)$. We prove that the local $v$-numbers of $J(H)<sup>t$ are linear in $t$ for all $t\ge1$. We further show that the global $v$-number of $J(H)<sup>t$ is linear in $t$ for all $t\ge n-1$. Finally, we prove that the global $v$-number of the powers of the cover ideal of any tree is linear in $t$ for all $t\ge1$.

• The paper proves that each local v-number of a power of a cover ideal of a unimodular hypergraph is exactly linear for every exponent, with slope determined by minimum-weight tight covers.
• It establishes global linearity from exponent t ≥ n−1 for all unimodular hypergraphs by combining total unimodularity, integer decomposition, and almost-cover constructions.
• For trees, the paper proves global linearity for every t ≥ 1 using recursive subtree invariants, while leaving the all-powers question open for general unimodular hypergraphs.

The $v$-number, introduced by Cooper, Seceleanu, Tohăneanu, Vaz Pinto, and Villarreal (2608.18406), measures the minimal degree of a form whose colon with an ideal yields a given associated prime. While Conca and, independently, Ficarra and Sgroi established eventual linearity of $v(I^t)$ in the exponent, the exact behavior for small powers remains delicate. This paper by Hang and Vu addresses that question for cover ideals of unimodular hypergraphs, extending earlier work of Vu on bipartite graphs.

Background and setup

Let $S = k[x_1,\ldots,x_n]$ be standard graded. For a hypergraph $H$ with edges $F_1,\ldots,F_m$, the cover ideal is

$J(H) = \bigcap_{i=1}^{m} (x_j \mid j \in F_i).$

$H$ is unimodular if its edge-vertex incidence matrix is totally unimodular; every bipartite graph is a unimodular hypergraph. Two facts anchor the paper: Herzog–Hibi–Trung proved that for unimodular $H$, ordinary and symbolic powers coincide, $J(H)^t = J(H)^{(t)}$ for all $t \ge 1$; and Chau–Ha–Jayanthan–Vu gave an integer-programming characterization of local $v(I^t)$0-numbers of symbolic powers. Specifically, for an associated prime $v(I^t)$1,

$v(I^t)$2

i.e., the minimum weight of a "$v(I^t)$3-almost $v(I^t)$4-cover." The paper's combinatorial vocabulary consists of $v(I^t)$5-covers, $v(I^t)$6-almost $v(I^t)$7-covers (weight $v(I^t)$8 on $v(I^t)$9, at least $S = k[x_1,\ldots,x_n]$0 elsewhere), and $S = k[x_1,\ldots,x_n]$1-tight $S = k[x_1,\ldots,x_n]$2-covers (exactly $S = k[x_1,\ldots,x_n]$3 on $S = k[x_1,\ldots,x_n]$4).

Local linearity via integer decomposition

The key structural input is the integer decomposition property of the covering polyhedron: for a unimodular hypergraph, every integral $S = k[x_1,\ldots,x_n]$5-cover decomposes as a sum of $S = k[x_1,\ldots,x_n]$6 integral 1-covers. The authors strengthen this to $S = k[x_1,\ldots,x_n]$7-almost covers via a clever gadget: they adjoin a new vertex $S = k[x_1,\ldots,x_n]$8 to the edge $S = k[x_1,\ldots,x_n]$9, producing a hypergraph $H$0 whose incidence matrix appends a standard basis column and hence stays totally unimodular. A $H$1-almost $H$2-cover $H$3 of $H$4 lifts to a $H$5-cover $H$6 of $H$7, which decomposes into $H$8 one-covers; exactly one contains the new vertex, and stripping it yields a decomposition $H$9 into a $F_1,\ldots,F_m$0-almost $F_1,\ldots,F_m$1-cover and a $F_1,\ldots,F_m$2-tight 1-cover.

This decomposition immediately gives the first main result:

Theorem 1. For a unimodular hypergraph $F_1,\ldots,F_m$3 and any edge $F_1,\ldots,F_m$4 of $F_1,\ldots,F_m$5,

$F_1,\ldots,F_m$6

for some constant $F_1,\ldots,F_m$7, for all $F_1,\ldots,F_m$8. Consequently, the global $F_1,\ldots,F_m$9-number $J(H) = \bigcap_{i=1}^{m} (x_j \mid j \in F_i).$0 is linear in $J(H) = \bigcap_{i=1}^{m} (x_j \mid j \in F_i).$1 for all $J(H) = \bigcap_{i=1}^{m} (x_j \mid j \in F_i).$2.

The proof shows the difference $J(H) = \bigcap_{i=1}^{m} (x_j \mid j \in F_i).$3 equals $J(H) = \bigcap_{i=1}^{m} (x_j \mid j \in F_i).$4, the minimum weight of a $J(H) = \bigcap_{i=1}^{m} (x_j \mid j \in F_i).$5-tight 1-cover, both from above (adding such a cover) and below (the decomposition lemma). For the global statement, the authors observe that $J(H) = \bigcap_{i=1}^{m} (x_j \mid j \in F_i).$6 and bound $J(H) = \bigcap_{i=1}^{m} (x_j \mid j \in F_i).$7 using the explicit almost-cover supported off $J(H) = \bigcap_{i=1}^{m} (x_j \mid j \in F_i).$8. Since distinct primes have slopes differing by integers at least 1 while constant terms differ by at most $J(H) = \bigcap_{i=1}^{m} (x_j \mid j \in F_i).$9, the prime achieving both minimal slope and minimal intercept dominates for $H$0.

This strictly generalizes Vu's theorem for bipartite graphs, and the linearity threshold $H$1 is uniform across all unimodular hypergraphs — a strong quantitative claim. Note, however, that the argument does not establish global linearity for all $H$2 in general; the gap between $H$3 and $H$4 persists for arbitrary unimodular hypergraphs.

Global linearity for trees

The second main result answers Vu's question affirmatively for trees:

Theorem 2. For any tree $H$5, $H$6 is linear in $H$7 for all $H$8.

By Vu's criterion, it suffices to find an edge $H$9 minimizing the local intercept $H$0 whose slope $H$1 equals $H$2. The proof develops a recursive calculus on rooted subtrees. Deleting an edge $H$3 splits $H$4 into components $H$5 and $H$6; define $H$7 and $H$8 as the minimum sizes of vertex covers of $H$9 excluding and including $J(H)^t = J(H)^{(t)}$0, respectively, and set $J(H)^t = J(H)^{(t)}$1. These quantities satisfy the recursion

$J(H)^t = J(H)^{(t)}$2

which follows from decomposing $J(H)^t = J(H)^{(t)}$3 into the subtrees $J(H)^t = J(H)^{(t)}$4 over neighbors $J(H)^t = J(H)^{(t)}$5.

The local data then take closed form:

• $J(H)^t = J(H)^{(t)}$6,
• $J(H)^t = J(H)^{(t)}$7,
• $J(H)^t = J(H)^{(t)}$8.

A further identity, $J(H)^t = J(H)^{(t)}$9 for two edges through $t \ge 1$0, drives the key claim: $t \ge 1$1 and $t \ge 1$2 cannot both be negative for an edge minimizing $t \ge 1$3. Indeed, since $t \ge 1$4 always, negativity of $t \ge 1$5 forces some neighbor $t \ge 1$6 with $t \ge 1$7, whence $t \ge 1$8, contradicting minimality. With $t \ge 1$9, the third term in the minimum defining $v(I^t)$00 is redundant, so $v(I^t)$01 and the global function is linear from $v(I^t)$02 onward.

Limitations and open questions

Two restrictions are worth stating plainly. First, the global linearity for general unimodular hypergraphs holds only for $v(I^t)$03; whether it holds for all $v(I^t)$04 — as it does for trees — remains open, and this is precisely Vu's original question in its full generality. Second, the tree argument exploits the acyclic structure essentially: the recursion for the $v(I^t)$05-invariants relies on unique paths, and no analogue is provided for graphs containing cycles beyond trees. Whether the techniques extend to unimodular graphs with cycles, or to broader classes of balanced hypergraphs, is not addressed.

Conclusion

The paper establishes that local $v(I^t)$06-numbers of powers of cover ideals of unimodular hypergraphs are exactly linear in the exponent for all $v(I^t)$07, with slope determined by minimum-weight tight covers, and proves global linearity for all powers when the hypergraph is a tree. The methods combine total unimodularity and integer decomposition with a subtree recursion, and they settle Vu's linearity question for trees while leaving its full scope for arbitrary unimodular hypergraphs unresolved.