Explicit classical-covariant expansion of Hilbert's power-detection covariants $C_\nu$ for binary forms, for arbitrary $\nu$

David Hilbert's four-page note of 1886 constructed, for a complex binary form $F$ of degree $n$ and each base degree $\nu$ (with $\mu=n/\nu$ the exponent), a polynomial covariant $C_\nu$ — of degree $\nu+1$ in the coefficients of $F$ and order $(n-2)(\nu+1)$ in the variables — whose identical vanishing is necessary and sufficient for $F$ to be the $\mu$-th power of a degree-$\nu$ binary form. Hilbert gave the general definition but wrote out explicit expansions in classical covariants only for $\nu=1,2,3,4$ ($C_1$ is the Hessian; $C_2$ the classical cubic covariant $T$ detecting powers of a quadratic). Modern work has resolved the surrounding structural questions: Abdesselam and Chipalkatti (2014) reconstructed the Hilbert covariant $H_{r,d}$ for all pairs $(r,d)$, proved that its coefficients define the variety of powers scheme-theoretically, and lifted the construction set-theoretically to forms in any number of variables by the Clebsch transfer principle; Raicu, Sam, Weyman and Yang (2026) determined the full homogeneous ideal of the variety of $a$-th powers of degree-$b$ binary forms — generated in degree $b+1$ with a linear minimal free resolution — confirming the conjecture of Abdesselam and Chipalkatti that the ideal is generated in degree $b+1$ (their Conjecture 5.1). What no work in this lineage supplies is the object at which the 1886 note stopped: an explicit universal expansion of $C_\nu$ in classical covariants (equivalently, iterated transvectants of $F$) for $\nu\ge 5$. The problem is to supply exactly that: closed-form expansions of the Hilbert covariant family, with coefficients that are explicit universal rational functions of the ground degree $n$, valid for arbitrary $\nu$.

Background

A binary form of degree $n$ (an $n$-ic) over $\mathbb{C}$ is a homogeneous polynomial $F(x_1,x_2)=\sum_{i=0}^{n}\binom{n}{i}a_i x_1^{n-i}x_2^{i}$; it is determined up to scale by its $n+1$ coefficients $a_i$, so forms of degree $n$ are parametrized by a projective space $\mathbb{P}^n$. Classical invariant theory studies the polynomial expressions in the $a_i$ (invariants) and in the $a_i$ together with the variables (covariants) that are compatible with the action of the group $\mathrm{SL}_2$ of linear changes of variables. A covariant of degree $m$ and order $q$ is a bihomogeneous polynomial $C(\mathbf{a},\mathbf{x})$, of degree $m$ in the coefficients and degree $q$ in $x_1,x_2$, transforming equivariantly; its weight is the number $w=(mn-q)/2$. The basic calculus for building covariants is the transvectant: for binary forms $A$ and $B$ of orders $p$ and $q$, the $k$-th transvectant $(A,B)_k$ is the image of $A\otimes B$ under the projection onto the irreducible $\mathrm{SL}_2$-summand $S^{p+q-2k}$ in the Clebsch–Gordan decomposition, given by an explicit constant-coefficient differential formula; for example the Hessian $\mathrm{He}(F)=(F,F)_2$ is a covariant of degree 2 and order $2n-4$.

A fundamental classical fact is that the Hessian of a binary $n$-ic vanishes identically if and only if $F$ is the $n$-th power of a linear form, i.e. $F=(p x_1+q x_2)^n$. In a four-page note published in 1886 in the Mathematische Annalen, the young David Hilbert asked and answered the general question behind this fact: when is a binary $n$-ic $F$ a perfect power $F=g^{\mu}$ of a form $g$ of degree $\nu$ (so $n=\mu\nu$)? He proved that there is a single covariant, which he called $C_\nu$, of degree $\nu+1$ and order $(n-2)(\nu+1)$, whose identical vanishing is necessary and sufficient for $F$ to be a $\mu$-th power of a degree-$\nu$ form. His construction starts from the elementary observation that $F^{1/\mu}$ is a polynomial of degree at most $\nu$ in a dehomogenized variable exactly when $F$ is a $\mu$-th power, converts the condition that the $(\nu+1)$-st derivative of $F^{1/\mu}$ vanish into an $\mathrm{SL}_2$-equivariant statement, and clears the fractional power. In modern notation (Proposition 4.3 of Abdesselam–Chipalkatti, reproduced in Ottaviani's survey), introduce auxiliary variables $y_1,y_2$ and set $D=x_1y_2-x_2y_1$; then

$$C_\nu \;=\; \frac{F(x_1,x_2)^{\,\nu+1-\frac{\nu}{n}}}{D^{\,\nu+1}}\left(y_1\frac{\partial}{\partial x_1}+y_2\frac{\partial}{\partial x_2}\right)^{\nu+1}\!\left[F(x_1,x_2)^{\,\frac{\nu}{n}}\right],$$

where the fractional powers are formal and the auxiliary variables cancel: the right-hand side expands to a genuine polynomial covariant of degree $\nu+1$, weight $\nu+1$ and order $(n-2)(\nu+1)$ in the variables $x_1,x_2$. Ottaviani's survey records the equivalent one-sided raising-operator form $C_\nu=x_1^{\,-2(\nu+1)}\,F^{\nu+1-\nu/n}\,\Delta^{\nu+1}(F^{\nu/n})$, where $\Delta=\sum_{i=0}^{n-1}(n-i)a_{i+1}\,\partial/\partial a_i$ is the classical weight-raising differential operator on the coefficients: the numerator $F^{\nu+1-\nu/n}\,\Delta^{\nu+1}(F^{\nu/n})$ is divisible by $x_1^{2(\nu+1)}$, and the quotient is the same covariant. When $\nu$ does not divide $n$, the same formula with $e=\gcd(\nu,n)$ detects $F$ being a $(n/e)$-th power of a degree-$e$ form. Hilbert computed the first cases explicitly in the classical covariants of the ground form — $C_1=\mathrm{He}(F)$, $C_2=(F,(F,F)_2)_1$, and $C_3$, $C_4$ as combinations such as $3(2n-3)\,[(F,F)_2]^2-2(n-2)\,F^2\,(F,F)_4$ — each formula valid for every ground degree $n$ with coefficients rational in $n$.

Geometrically, the coefficients of $C_\nu$ are defining equations for the variety $X_{(a^b)}\subset\mathbb{P}^{ab}$ of $a$-th powers of degree-$b$ binary forms (with $a=\mu$, $b=\nu$), a special 'coincidence root locus' of forms whose roots all have multiplicity divisible by $a$, and the note's criterion says these equations cut out $X_{(a^b)}$ set-theoretically. The note itself posed no explicit open question, but it visibly stopped short of carrying its program past $\nu=4$: the general definition of $C_\nu$ is given, while the expansion of $C_\nu$ into expressions built from known covariants of the ground form is carried out only for $\nu=1,2,3,4$.

The paper was then forgotten for about a century (a short 1896 report by Brioschi aside), until Rota's 1999 account of turning points in invariant theory advertised Hilbert's forgotten solution. The modern revival is due to Abdesselam and Chipalkatti: in a 2014 paper they defined the Hilbert covariant $H_{r,d}$ for arbitrary $(r,d)$, proved it coincides up to a scalar with a Wronskian construction, proved that when $r\mid d$ the coefficients of $H_{r,d}$ define the variety of powers scheme-theoretically (their saturation is the full homogeneous ideal), and extended the construction to forms in any number of variables via the classical Clebsch transfer principle (the classical procedure of Clebsch that lifts binary covariants to concomitants — the several-variable analogue of covariants — of forms in more variables). In 2026, Raicu, Sam, Weyman and Yang completed the structural picture for binary forms: the homogeneous ideal of $X_{(a^b)}$ is generated in degree $b+1$ by the maximal minors of an explicit matrix of linear forms — equivalently, by the coefficients of the family of degree-$(b+1)$ 'Göttingen covariants' of Abdesselam–Chipalkatti vanishing on the variety, among which Hilbert's is the single one of top order — and has a linear minimal free resolution, confirming the Abdesselam–Chipalkatti conjecture (their Conjecture 5.1). They work representation-theoretically, through a derived analogue of Hermite reciprocity for $\mathrm{SL}_2$-representations (Hermite reciprocity being the classical isomorphism $\mathrm{Sym}^m(\mathrm{Sym}^n\mathbb{C}^2)\cong\mathrm{Sym}^n(\mathrm{Sym}^m\mathbb{C}^2)$, and plethysm the general decomposition of symmetric powers of symmetric powers). Notably, the explicit classical-covariant expansions also stop at $\nu=4$ in the modern literature: Abdesselam and Chipalkatti write out only the cases $r\le 4$ and remark that their symbolic recipe 'quickly begins to get more and more tedious' as $r$ grows.

Problem Statement

For the generic complex binary form $F$ of degree $n$, and each base degree $\nu$ (with exponent $\mu=n/\nu$ when $\nu\mid n$; when $\nu\nmid n$, the same formula with $e=\gcd(\nu,n)$ defines a covariant detecting $(n/e)$-th powers of degree-$e$ forms), the Hilbert covariant is the polynomial covariant of degree $\nu+1$ in the coefficients and order $(n-2)(\nu+1)$ in the variables, determined up to a nonzero scalar — for $\nu\mid n$, uniquely among the covariants of its type vanishing on the variety of $\mu$-th powers — by the closed formula

$$C_\nu \;=\; \frac{F^{\,\nu+1-\frac{\nu}{n}}}{(x_1y_2-x_2y_1)^{\,\nu+1}}\left(y_1\frac{\partial}{\partial x_1}+y_2\frac{\partial}{\partial x_2}\right)^{\nu+1}\!\left[F^{\,\frac{\nu}{n}}\right],$$

in which the fractional powers are formal and the auxiliary variables $y_1,y_2$ cancel, and whose identical vanishing is equivalent to $F$ being a $\mu$-th power of a degree-$\nu$ form. Hilbert (1886) — and, reproducing him, Abdesselam–Chipalkatti (2014) — wrote out explicit expansions of $C_\nu$ in classical covariants of the ground form only for $\nu=1,2,3,4$: $C_1=(F,F)_2$ (the Hessian), $C_2=(F,(F,F)_2)_1$ (the classical cubic covariant $T$, which detects powers of a quadratic), $C_3=3(2n-3)[(F,F)_2]^2-2(n-2)F^2(F,F)_4$, and an analogous degree-5 expression for $C_4$, each up to a harmless scalar, valid for every ground degree $n$ with coefficients rational in $n$.

The problem is to produce, for arbitrary $\nu$ — in particular for every $\nu\ge 5$ — an explicit expansion of $C_\nu$ as a finite linear combination of monomials in the iterated transvectants of $F$ with itself, with coefficients given by explicit universal rational functions of the ground degree $n$. The ground degree $n$ ranges over the admissible values — that is, each $n$ with $n\nmid\nu$, so that the detected power exponent ($\mu=n/\nu$ when $\nu\mid n$, and $n/e$ with $e=\gcd(\nu,n)$ otherwise) is at least $2$; when $n\mid\nu$ the variety of powers is all of $\mathbb{P}^n$ and $C_\nu$ vanishes identically (Abdesselam–Chipalkatti 2014, §3.5). The covariant is determined up to a nonzero scalar: the equivalent-vanishing property established by Hilbert is shared by any expression proportional to the defining formula above (or to its Wronskian equivalent), and for $\nu\mid n$ the covariants of degree $\nu+1$ and order $(n-2)(\nu+1)$ vanishing on the variety of powers form a one-dimensional space spanned by $C_\nu$ — the degree-$(\nu+1)$ part of the ideal being the span of the family of Göttingen covariants, whose top-order member, corresponding to a Cartan component (which occurs with multiplicity one), is $C_\nu$ itself (Raicu–Sam–Weyman–Yang 2026, confirming Abdesselam–Chipalkatti's Conjecture 5.1). The problem is deliberately restricted to binary forms, matching the scope of the 1886 source; the extension to forms in more than two variables has been treated set-theoretically by the Clebsch transfer principle and is not part of this question.

References

1. David Hilbert, Ueber die nothwendigen und hinreichenden covarianten Bedingungen für die Darstellbarkeit einer binären Form als vollständiger Potenz, Mathematische Annalen 27 (1886), no. 1, 158–161, DOI 10.1007/BF01447309, https://doi.org/10.1007/BF01447309
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3. F. Brioschi, Sopra un Teorema del Sigr. Hilbert, Rendiconti del Circolo Matematico di Palermo 10 (1896), 153–157, DOI 10.1007/bf03021461, https://doi.org/10.1007/bf03021461
4. Gian-Carlo Rota, Two turning points in invariant theory, The Mathematical Intelligencer 21 (1999), no. 1, 20–27, DOI 10.1007/bf03024826, https://doi.org/10.1007/bf03024826
5. Abdelmalek Abdesselam and Jaydeep Chipalkatti, On the Wronskian combinants of binary forms, Journal of Pure and Applied Algebra 210 (2007), no. 1, 43–61, DOI 10.1016/j.jpaa.2006.08.006, https://arxiv.org/abs/math/0507488
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7. Carlos D'Andrea, Jaydeep Chipalkatti, and Abdelmalek Abdesselam, On the Jacobian ideal of the binary discriminant, arXiv:math/0601705 (2006), with an appendix by A. Abdesselam, https://arxiv.org/abs/math/0601705
8. Abdelmalek Abdesselam and Jaydeep Chipalkatti, On Hilbert covariants, Canadian Journal of Mathematics 66 (2014), no. 1, 3–30, DOI 10.4153/CJM-2012-046-1, https://doi.org/10.4153/CJM-2012-046-1
9. Claudiu Raicu, Steven V Sam, Jerzy Weyman, and Fuxiang Yang, Powers of binary forms and derived Hermite reciprocity, arXiv:2602.15175 (2026), https://arxiv.org/abs/2602.15175
10. Giorgio Ottaviani, with an appendix by Vincenzo Galgano, Some Classical Invariants, from Harmonic Quadruples to Triangle Groups, arXiv:2603.03413 (2026), https://arxiv.org/abs/2603.03413