The v-numbers of permanental ideals

Abstract: In this article, we compute the $\vv$-number of $2\times 2$ permanental ideals of generic, generic symmetric, and generic Hankel matrices.

• The paper completely computes the v-number of 2×2 permanental ideals for generic, generic symmetric, and generic Hankel matrices over fields of characteristic not 2, with all values in {0, 1, 2, 3}.
• The authors combine colon ideal identities, Gröbner-basis calculations, degree bounds, and retraction arguments to establish matching upper and lower bounds for each matrix class.
• The results show strong structural dependence: generic and symmetric cases typically stabilize at 3, while Hankel ideals can have v-number 1 for sufficiently large matrices; characteristic 2 yields the trivial value 0 because permanental and determinantal ideals coincide.

Overview

This paper computes the $v$-number of $2\times 2$ permanental ideals $P_2(X)$ for three classes of matrices over a field $K$ with $\operatorname{char} K \neq 2$: generic, generic symmetric, and generic Hankel matrices. The $v$-number, introduced by Cooper et al. (2605.11621), is defined as

$$v(I) = \min\{d \mid \exists\ f \in R_d \text{ and } p \in \operatorname{Ass}(I) \text{ such that } I : f = p\},$$

and was originally motivated by the study of minimum distance functions of projective Reed–Muller-type codes. The paper's main results are complete and uniform: the value depends only on matrix size and type, taking values in $\{0, 1, 2, 3\}$.

Methodology

The proofs rest on two standard bounds from Grisalde–Reyes–Villarreal (2605.11621). For an upper bound, if $(I : f)$ is prime for a homogeneous polynomial $f$, then $2\times 2$0. For a lower bound,

$2\times 2$1

where $2\times 2$2 denotes the least degree of a homogeneous element of $2\times 2$3. The lower bound is established by showing that certain colon ideals contain no forms below a given degree; this is done via Gröbner bases and the observation that $2\times 2$4 implies $2\times 2$5. The colon ideal computations themselves use Laubenbacher–Swanson's Gröbner basis for permanental ideals of generic matrices, Grieco–Guerrieri–Swanson's basis for the Hankel case, and Chau's basis for the symmetric case, together with a lemma guaranteeing that a $2\times 2$6-primary ideal $2\times 2$7 containing $2\times 2$8 with $2\times 2$9 and $P_2(X)$0 must equal $P_2(X)$1.

In characteristic 2, permanental ideals coincide with determinantal ideals and are prime, so $P_2(X)$2 trivially; the interesting content is entirely in odd characteristic. The $P_2(X)$3 case is also excluded since $P_2(X)$4 is prime there in all three settings.

Generic matrices

For a generic $P_2(X)$5 matrix $P_2(X)$6 with $P_2(X)$7, the results are:

Case	$P_2(X)$8
$P_2(X)$9, $K$0	2
$K$1	3

When $K$2, the upper bound comes from showing that $K$3 is an associated prime equal to $K$4, giving $K$5. The matching lower bound shows that no colon ideal $K$6 contains a linear form, using the initial ideal computation $K$7 and a specialization argument forcing all coefficients to vanish.

For $K$8, the upper bound uses the colon identity $K$9, which is prime. The lower bound is the most substantial combinatorial argument in this section: any quadratic form lying in $\operatorname{char} K \neq 2$0 is shown to lie in $\operatorname{char} K \neq 2$1 itself, via coefficient elimination under specializations and a final contradiction against the known initial ideal. Since every associated prime contains either a full row or a full column of variables, this bounds $\operatorname{char} K \neq 2$2 uniformly.

Generic symmetric matrices

For an $\operatorname{char} K \neq 2$3 generic symmetric matrix $\operatorname{char} K \neq 2$4 with $\operatorname{char} K \neq 2$5, the paper proves $\operatorname{char} K \neq 2$6. The minimal primes are exactly the ideals $\operatorname{char} K \neq 2$7, so it suffices to show $\operatorname{char} K \neq 2$8 for all pairs. This is proved by induction on $\operatorname{char} K \neq 2$9: the base case $v$0 is handled by explicit reduction against the Gröbner basis, and the inductive step relies on a retract lemma showing that the natural map $v$1 splits, where $v$2 is obtained by deleting the first row and column — hence $v$3. A key claim in the induction is that the leading monomial of a hypothetical quadratic element cannot involve first-row variables; otherwise, specializing to a $v$4 principal submatrix contradicts the base case. The upper bound follows from the known associated prime $v$5.

Generic Hankel matrices

The Hankel case exhibits the richest behavior, with the answer depending delicately on $v$6:

Case	$v$7
$v$8, $v$9; or $$v(I) = \min\{d \mid \exists\ f \in R_d \text{ and } p \in \operatorname{Ass}(I) \text{ such that } I : f = p\},$$0	1
$$v(I) = \min\{d \mid \exists\ f \in R_d \text{ and } p \in \operatorname{Ass}(I) \text{ such that } I : f = p\},$$1, $$v(I) = \min\{d \mid \exists\ f \in R_d \text{ and } p \in \operatorname{Ass}(I) \text{ such that } I : f = p\},$$2; or $$v(I) = \min\{d \mid \exists\ f \in R_d \text{ and } p \in \operatorname{Ass}(I) \text{ such that } I : f = p\},$$3	2
$$v(I) = \min\{d \mid \exists\ f \in R_d \text{ and } p \in \operatorname{Ass}(I) \text{ such that } I : f = p\},$$4	3

The $$v(I) = \min\{d \mid \exists\ f \in R_d \text{ and } p \in \operatorname{Ass}(I) \text{ such that } I : f = p\},$$5 cases exploit the striking colon identity $$v(I) = \min\{d \mid \exists\ f \in R_d \text{ and } p \in \operatorname{Ass}(I) \text{ such that } I : f = p\},$$6 from Grieco–Guerrieri–Swanson when $$v(I) = \min\{d \mid \exists\ f \in R_d \text{ and } p \in \operatorname{Ass}(I) \text{ such that } I : f = p\},$$7: a single variable suffices to reach the maximal ideal as an associated prime. For $$v(I) = \min\{d \mid \exists\ f \in R_d \text{ and } p \in \operatorname{Ass}(I) \text{ such that } I : f = p\},$$8, the authors verify $$v(I) = \min\{d \mid \exists\ f \in R_d \text{ and } p \in \operatorname{Ass}(I) \text{ such that } I : f = p\},$$9 directly, exhibiting each generator $\{0, 1, 2, 3\}$0 as a signed combination of subpermanents and ruling out proper containment via Gröbner reduction of $\{0, 1, 2, 3\}$1. Notably, the $\{0, 1, 2, 3\}$2 case is resolved by a Macaulay2 verification that $\{0, 1, 2, 3\}$3 equals the $\{0, 1, 2, 3\}$4 ideal over $\{0, 1, 2, 3\}$5, hence over every field — an identification that is not structurally explained within the paper.

The $\{0, 1, 2, 3\}$6 cases follow the same template: explicit colon identities give the upper bound (e.g., $\{0, 1, 2, 3\}$7 for $\{0, 1, 2, 3\}$8), while the lower bound reduces to showing that specific colon ideals contain no linear forms, proved by iterative Gröbner reductions. The two smallest cases, $\{0, 1, 2, 3\}$9 and $(I : f)$0, require degree-2 elimination arguments analogous to the generic-matrix lemma.

Limitations and open questions

Several aspects of the paper warrant note. First, all results assume $(I : f)$1; in characteristic 2 the ideals are prime and the invariant is trivially zero, so no characteristic-uniform behavior beyond this dichotomy is established. Second, the analysis is restricted to $(I : f)$2; the $(I : f)$3-number of $(I : f)$4 for $(I : f)$5 remains open for all three matrix classes, and the methods here rely heavily on the explicit small-$(I : f)$6 Gröbner bases available in the literature, so extension is not immediate. Third, the classification of minimal primes of permanental ideals is incomplete in general, which constrains the applicability of the lower-bound technique to larger $(I : f)$7. Finally, the equality between the $(I : f)$8 and $(I : f)$9 Hankel permanental ideals is established only computationally, leaving open a structural explanation.

Conclusion

The paper settles the $f$0-number of $f$1 permanental ideals of generic, generic symmetric, and generic Hankel matrices completely, showing values bounded above by 3 across all cases. The results demonstrate that the invariant is sensitive to matrix structure: it stabilizes at 3 for large generic and symmetric matrices, but drops to 1 for sufficiently large Hankel matrices due to the existence of single-variable colon identities reaching the maximal ideal. The techniques — combining colon ideal computations, Gröbner-based degree elimination, and retraction arguments — provide a template likely applicable to related binomial-permanent settings, though the $f$2 case remains unresolved.