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Hasse-Witt invariants for trace forms of Jacobi polynomials

This paper revisits and generalizes the calculation of the Hasse-Witt invariant for trace forms of Generalized Laguerre Polynomials by providing an alternate combinatorial derivation of the underlying determinant and applying these techniques to the broader family of Jacobi polynomials.

Original authors: John Cullinan, Farshid Hajir, Elisabeth Young

Published 2026-08-13

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CC BY 4.0

Original authors: John Cullinan, Farshid Hajir, Elisabeth Young

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where numbers aren't just for counting apples or calculating change, but are the secret ingredients in a cosmic recipe for building entire universes of mathematics. This is the realm of Galois theory, a branch of math that studies how equations can be "unlocked" to reveal their hidden symmetries. Think of a polynomial equation like a locked treasure chest. The "roots" are the treasures inside, and the "Galois group" is the specific set of keys (or symmetries) that can shuffle those roots around without breaking the chest's rules. Sometimes, mathematicians want to know if they can build a bigger, more complex chest (a larger field) that contains the first one, but with a specific, tricky new lock (a larger group). This is called the "group extension problem." It's like asking: "If I have a simple puzzle, can I build a bigger puzzle that fits perfectly inside a specific, complicated frame?"

To solve these riddles, mathematicians use a special tool called the "Hasse-Witt invariant." Imagine this as a magical litmus test or a security scanner. When you run a number field through this scanner, it spits out a simple "yes" or "no" (represented by the numbers 1 or -1) to tell you if your big puzzle can actually fit into the frame you want. If the scanner says "yes" (1) for every possible angle of light (every prime number), then the extension exists. If it says "no" (-1) even once, the dream is impossible. For decades, experts could only run this scanner on a very specific, narrow type of puzzle called "Generalized Laguerre Polynomials." They had a manual for how to calculate the test, but it was written in a secret code that only worked for that one specific puzzle type.

This paper is about cracking that code and building a universal scanner. The authors, John Cullinan, Farshid Hajir, and Elisabeth Young, take the old, clunky manual and rewrite it using a fresh, powerful method based on "Hankel determinants"—which you can think of as a special kind of pattern-matching grid. They apply this new method to a much broader family of puzzles called "Jacobi Polynomials," which are like a giant, two-parameter toolbox containing the old Laguerre puzzles as just one small special case. By doing this, they don't just solve the problem for one puzzle; they give us a master key that works for an infinite variety of them.

Here is what they found: They successfully calculated the exact formula for the "Hasse-Witt invariant" for these Jacobi Polynomials. In plain terms, they figured out the precise recipe to run the security scanner on this entire new family of equations. They proved that for any Jacobi polynomial (defined by two numbers, $\alpha$ and $\beta$ ), you can now explicitly calculate whether it can be embedded into a larger, more complex Galois extension. Their work confirms that the old, specific method used by a mathematician named Feit was correct, but it also shows that Feit's method was just a tiny slice of a much larger pie. By using their new combinatorial techniques (which are like clever counting tricks), they derived a new, explicit formula for the determinant $\Delta_t$ (the core number needed for the test) that works for the Jacobi family.

The paper explicitly rules out the idea that you need special, unique properties of the Laguerre polynomials to solve this problem. Instead, they show that the solution relies on general patterns found in the "power-sums" of the roots, which can be organized into a Hankel matrix. They don't just suggest this works; they provide a rigorous mathematical proof that their new formula is equivalent to the old one for the Laguerre case and extends it perfectly to the Jacobi case. They even provide concrete examples: for a specific set of numbers ( $n=5, \alpha=-18, \beta=24$ ), their scanner says "no" (the extension doesn't exist), but for another set ( $n=5, \alpha=-29/2, \beta=5$ ), it says "yes" (the extension exists). This proves their method isn't just theoretical; it can distinguish between solvable and unsolvable cases in real time.

The beauty of their discovery is that it turns a problem that required a unique, ad-hoc calculation for every new type of polynomial into a standard procedure. They showed that by looking at the "generating function" (a fancy way of summarizing the sequence of numbers) as a "continued fraction" (a nested fraction structure), they could read off the necessary numbers directly. This means that for any future mathematician working with these polynomials, the heavy lifting of calculating the Hasse-Witt invariant is now done. They have provided the map, the compass, and the formula, allowing anyone to navigate these complex algebraic landscapes and determine exactly which "group extensions" are possible and which are impossible, all without needing to reinvent the wheel for every new puzzle.

Technical Summary: Hasse-Witt Invariants for Trace Forms of Jacobi Polynomials

Problem Statement
The paper addresses the group extension problem within Galois theory, specifically the conditions under which a Galois extension $L/\mathbb{Q}$ with group $B$ can be embedded into a larger extension $M/\mathbb{Q}$ with group $G$ , where $G$ is a non-split central extension of $B$ by $C_2$ . Following the work of Feit [1] and Hajir [2], the solvability of this embedding problem for specific groups (such as $\tilde{A}_n$ ) relies on the Hasse-Witt invariant of the trace form associated with the root field of a defining polynomial.

The core computational obstacle is the explicit determination of the Hasse-Witt invariant, which requires calculating the diagonalization coefficients ( $D_t$ ) of the trace form. These coefficients are derived from the principal minors ( $\Delta_t$ ) of the power-sum matrix of the polynomial's roots. While Feit successfully computed $\Delta_t$ for Generalized Laguerre Polynomials (GLP) using properties specific to that family (recursion relations and congruences), these techniques do not generalize to broader families of orthogonal polynomials. Consequently, extending the existence results for $\tilde{A}_n$ -extensions beyond the specific GLP cases required a more general method for computing $\Delta_t$ .

Methodology
The authors propose a shift in methodology from polynomial-specific properties to a combinatorial approach based on Hankel determinants. The strategy involves:

Reformulation as Hankel Determinants: The authors identify $\Delta_t$ as the $t \times t$ principal minor of a Hankel matrix formed by the power-sums of the roots.
Generating Functions and Continued Fractions: They utilize the generating function $G(z)$ for the power-sums of the roots. By expressing $G(z)$ as a continued fraction (specifically a Stieltjes-type continued fraction), they leverage a known theorem (referenced as [3, Thm. 30]) that relates the coefficients of the continued fraction directly to the Hankel determinants.
Recursive Coefficient Calculation: For the Jacobi polynomials $J^{(\alpha, \beta)}_n(x)$ , the authors derive the coefficients ( $a_i$ ) of the continued fraction expansion of $G(z)$ . This involves defining a sequence of auxiliary functions $f_i(m)$ and establishing recursive relations (Proposition 30) to simplify the calculation of these coefficients.
Explicit Formula Derivation: Using the calculated coefficients, the authors derive an explicit product formula for $\Delta_t$ . This formula is then used to compute the diagonalization coefficients $D_t = \Delta_{t+1}/\Delta_t$ and subsequently the Hasse-Witt invariant $\epsilon_p(E)$ .

Key Contributions and Results
The paper provides the following specific results:

Explicit Formula for $\Delta_t$ : Theorem 13 presents a closed-form expression for the principal minors $\Delta_t$ of the power-sum matrix for the monic Jacobi polynomials $J^{(\alpha, \beta)}_n(x)$ . The formula is given as $\Delta_t = A_t B_t / C_t$ , where $A_t, B_t,$ and $C_t$ are defined by specific products involving the parameters $\alpha, \beta$ and the degree $n$ (Equations 10–12).
Hasse-Witt Invariant for Jacobi Polynomials: By combining the explicit $\Delta_t$ formula with Serre's theorem and properties of the Hilbert symbol, the authors derive an explicit expression for the Hasse-Witt invariant $\epsilon_p(E)$ for the root fields of Jacobi polynomials (Section 4).
Generalization of Feit's Results: The paper demonstrates that the Generalized Laguerre Polynomials are a limiting case of the Jacobi polynomials. By applying their general method to the specific parameters corresponding to GLPs, the authors recover Feit's original formula for $\Delta_t$ (Section 4.1), thereby providing an alternative, combinatorial derivation of Feit's theorem.
Concrete Examples: The authors provide explicit numerical examples for $n=5$ with specific parameters $(\alpha, \beta)$ . They show cases where the Hasse-Witt invariant is non-trivial (preventing the embedding into $\tilde{A}_5$ ) and cases where it is trivial (allowing the embedding), illustrating the practical application of their formulas.

Significance and Claims
The authors claim that their primary contribution is situating the calculation of $\Delta_t$ within the context of Hankel determinants and continued fractions. They assert that:

The techniques used are not specific to Jacobi polynomials but are widely applicable to any sequence of polynomials indexed by degree.
The method does not rely on orthogonality, distinguishing it from traditional approaches that might depend on specific weight functions or orthogonality relations.
The introduction of an additional parameter (moving from one-parameter GLPs to two-parameter Jacobi polynomials) implies that for each $n \geq 2$ , there may be infinitely many solutions to the group extension problem for $\tilde{A}_n$ , thereby generalizing the finiteness constraints of previous constructions.

The paper concludes by noting that while the explicit formula for $\Delta_t$ in [1] and [2] relied on properties that did not generalize, the combinatorial approach presented here successfully extends these results to the broader family of Jacobi polynomials.

Technical Summary: Hasse-Witt Invariants for Trace Forms of Jacobi Polynomials

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