Hankel Determinantal Ring have $F$-regular Singularities

Abstract: In this article, we study Hankel determinantal rings, a special class of determinantal rings defined by minors of Hankel matrices of indeterminates. We discuss the question posed in \cite[Question~4.8]{conca2018hankel} by showing that the test ideal of a Hankel determinantal ring coincides with the ring itself.

• The paper proves that the test ideal equals the entire ring, establishing strong F-regularity for square Hankel determinantal hypersurfaces in every prime characteristic without the prior restriction p ≥ n.
• The authors use Fedder-type trace maps and the explicit monomial structure of Hankel determinants to reduce test-ideal elements by their central-variable exponents until they obtain 1.
• The result strengthens earlier F-rationality theorems and leaves strong F-regularity for non-maximal Hankel minors as a significant open problem, supported only by limited computations.

Overview

This paper, by Jyoti Singh and Nikhil P. Zade (2608.17435), resolves a case of a question posed by Conca, Mostafazadehfard, Singh, and Varbaro concerning the $F$-regularity of Hankel determinantal rings. The main theorem states that if $S = K[x_1,\dots,x_{2n-1}]$ is a polynomial ring over a perfect field $K$ of positive characteristic $p$, and $f$ is the determinant of the $n \times n$ Hankel matrix $H$ with $(i,j)$-entry $x_{i+j-1}$, then the test ideal of the hypersurface ring $R := S/(f)$ equals $S = K[x_1,\dots,x_{2n-1}]$0 itself. Via Schwede's criterion — a domain essentially of finite type over a perfect field is strongly $S = K[x_1,\dots,x_{2n-1}]$1-regular if and only if its test ideal is the unit ideal — this establishes that Hankel determinantal rings defined by maximal minors of square Hankel matrices are strongly $S = K[x_1,\dots,x_{2n-1}]$2-regular for every prime characteristic, removing the restriction $S = K[x_1,\dots,x_{2n-1}]$3 present in prior work.

Background: test ideals and strong $S = K[x_1,\dots,x_{2n-1}]$4-regularity

The paper works within the framework of $S = K[x_1,\dots,x_{2n-1}]$5-singularities, where singularity severity in characteristic $S = K[x_1,\dots,x_{2n-1}]$6 is measured through the Frobenius endomorphism $S = K[x_1,\dots,x_{2n-1}]$7. Following Schwede rather than Hochster–Huneke's original tight closure formulation, the authors define the test ideal $S = K[x_1,\dots,x_{2n-1}]$8 of a reduced $S = K[x_1,\dots,x_{2n-1}]$9-finite Noetherian ring as the smallest nonzero uniformly $K$0-compatible ideal not contained in any minimal prime, where an ideal $K$1 is $K$2-compatible for $K$3 if $K$4. This Cartier-algebraic perspective connects directly to Frobenius splitting theory: uniformly compatible ideals are precisely those preserved by all Cartier maps, and their compatibly split subschemes include the non-strongly-$K$5-regular locus.

The key structural input is Fedder-type machinery for quotients of polynomial rings. For a homogeneous principal ideal $K$6 with $K$7, every $K$8-linear map $K$9 has the form $p$0 for some $p$1, where $p$2 is the trace map picking out the coefficient of $p$3.

Prior results on determinantal rings

Generic determinantal rings, Pfaffian rings, ladder determinantal rings, and symmetric determinantal rings are known to be strongly $p$4-regular. For Hankel determinantal rings $p$5, Conca et al. established $p$6-purity in general and $p$7-rationality for maximal minors when $p$8. Since hypersurfaces are Gorenstein, $p$9-rationality implies strong $f$0-regularity there, but only under the characteristic bound $f$1. The question of $f$2-regularity in arbitrary characteristic remained open; the present paper answers it affirmatively in the maximal-minor square case.

The main theorem and its proof

The proof exploits the explicit monomial structure of the Hankel determinant. Writing $f$3 for the monomial $f$4, one has $f$5 in $f$6. The strategy is constructive: given any nonzero element $f$7, expressed as a $f$8-linear combination of monomials with $f$9-exponent less than $n \times n$0, choose $n \times n$1 large enough that $n \times n$2, and select a multiplier $n \times n$3 so that the composition $n \times n$4 sends $n \times n$5 to a simpler element still lying in $n \times n$6.

The argument proceeds by degree reduction on the $n \times n$7-exponent. Because $n \times n$8 contains both the monomial $n \times n$9 and the "checkerboard" monomial $H$0, careful choices of $H$1 isolate terms and produce elements $H$2, then $H$3, and so on, with each remainder $H$4 having strictly smaller $H$5-degree and vanishing exponents on progressively more variables. The parity of $H$6 determines the terminal step:

• Odd $H$7: the process terminates at $H$8, after which a suitable choice of $H$9 yields $(i,j)$0, and a final trace computation gives $(i,j)$1.
• Even $(i,j)$2: the process terminates at $(i,j)$3, from which $(i,j)$4 follows (the cross-term contribution $(i,j)$5 vanishes by an exponent-counting argument), again yielding $(i,j)$6.

Since $(i,j)$7 contains $(i,j)$8, it equals $(i,j)$9, and Theorem 3.19 of Schwede–Tucker's survey gives strong $x_{i+j-1}$0-regularity. An immediate consequence is that these rings are $x_{i+j-1}$1-rational and have rational-type singularities in all characteristics, strengthening the $x_{i+j-1}$2 hypothesis of Conca et al.'s $x_{i+j-1}$3-rationality result to no hypothesis at all in the square maximal-minor case.

Supporting evidence beyond the theorem

The paper supplements the main result with two observations covering additional cases. First, a Hankel determinantal ring defined by a $x_{i+j-1}$4 Hankel matrix is isomorphic to the $x_{i+j-1}$5-th Veronese subring of $x_{i+j-1}$6, hence a direct summand of a regular ring; by Hochster–Huneke's direct summand theorem it is strongly $x_{i+j-1}$7-regular in positive characteristic. Second, a Macaulay2 computation verifies that the $x_{i+j-1}$8 Hankel determinantal ring over $x_{i+j-1}$9 cut out by all $R := S/(f)$0 minors has test ideal equal to the unit ideal, i.e., is strongly $R := S/(f)$1-regular at characteristic $R := S/(f)$2 — a case outside both the main theorem ($R := S/(f)$3) and the Veronese identification. These cases motivate the paper's closing question: whether every Hankel determinantal ring is strongly $R := S/(f)$4-regular.

Limitations and open questions

The method has a clear boundary acknowledged explicitly by the authors: the proof relies on the colon-ideal description of $R := S/(f)$5, which is fully explicit only when $R := S/(f)$6 is principal. When $R := S/(f)$7 is generated by multiple minors ($R := S/(f)$8), computing the module

$R := S/(f)$9

becomes substantially harder, and the paper offers no technique for this case beyond the computational example. Consequently, the general question — strong $S = K[x_1,\dots,x_{2n-1}]$00-regularity of $S = K[x_1,\dots,x_{2n-1}]$01 for arbitrary $S = K[x_1,\dots,x_{2n-1}]$02 — remains open, as does the corresponding question over non-perfect fields. The Macaulay2 verification at small parameters constitutes evidence but not a proof for the non-maximal minor case.

Conclusion

The paper establishes that Hankel determinantal hypersurfaces defined by maximal minors of square Hankel matrices are strongly $S = K[x_1,\dots,x_{2n-1}]$03-regular over every perfect field of positive characteristic, answering Question 4.8 of Conca et al. in this case. The proof is a direct, elementary trace-map computation exploiting the anti-diagonal structure of the Hankel determinant, avoiding tight closure entirely. The principal open problem left by the work is the extension to Hankel determinantal rings defined by non-maximal minors, where the Fedder-type description of Cartier maps is no longer available in closed form.