A pencil of binary forms is a two-dimensional linear system {λφ+μψ}\{\lambda\varphi+\mu\psi\} of homogeneous polynomials in two variables. Hilbert (1887) asked for all such pencils of order ν\nu whose third transvectant (φ,ψ)3(\varphi,\psi)_3 — the first pencil-combinant beyond the Jacobian — vanishes identically, up to linear change of the variables. He classified them completely for orders ν=3,…,8\nu=3,\dots,8 (with ν=6\nu=6 exceptional, carrying a two-parameter family), gave two explicit pencils valid for every order, and stated that the general enumeration was probably incomplete. A companion divisibility question from the same paper was answered affirmatively by A. Hirsch (Königsberg dissertation, 1892), as attested by Hilbert's own footnote and by F. Meyer's contemporary report on invariant theory. Later literature — Hilbert's 1888 solution of the prescribed-Jacobian problem, the Italian school's study of vanishing self-transvectants of single binary forms, and the modern combinants program of Abdesselam and Chipalkatti — leaves the general-order classification untouched. The surviving problem is to determine, for every order ν\nu, all pencils with identically vanishing third transvectant, completing or delimiting Hilbert's two families.

Background

A binary form of order ν\nu (a binary ν\nu-ic) is a homogeneous polynomial f(x1,x2)f(x_1,x_2) of degree ν\nu with complex coefficients; the group SL2SL_2 of 2×22\times 2 complex matrices of determinant one acts on the ν+1\nu+1 dimensional space of such forms by linear change of the variables, and classical invariant theory studies the quantities and forms that are invariant or equivariant under this action. The kk-th transvectant of two binary forms ff and gg, written (f,g)k(f,g)_k and called Ueberschiebung (overshift) in the older German literature, is the differential contraction

(f,g)k=∑i=0k(−1)i(ki)∂kf∂x1k−i∂x2i∂kg∂x1i∂x2k−i,(f,g)_k=\sum_{i=0}^{k}(-1)^i\binom{k}{i}\frac{\partial^k f}{\partial x_1^{k-i}\partial x_2^{i}}\frac{\partial^k g}{\partial x_1^{i}\partial x_2^{k-i}},

which is again a binary form; for k=1k=1 it is the Jacobian covariant of the pair. A pencil of binary forms is a two-dimensional linear system {λφ+μψ}\{\lambda\varphi+\mu\psi\}, and a combinant is a joint covariant of the two generators that depends only on the pencil, that is, is unchanged up to a power of the determinant of the change of basis when the generators are replaced by two independent linear combinations of themselves. The odd transvectants (φ,ψ)2r−1(\varphi,\psi)_{2r-1} are relative combinants of the pencil of weight one: each acquires a factor equal to the determinant of the change of basis, so that in particular their vanishing depends only on the pencil; the first, (φ,ψ)1(\varphi,\psi)_1, is its Jacobian, and the third, (φ,ψ)3(\varphi,\psi)_3, is the next one.

Hilbert's 1887 paper determines the pencils whose third transvectant vanishes identically, a binary form of order 2ν−62\nu-6 with 2ν−52\nu-5 coefficients. The parameter count makes the problem sharp: the pencil carries 2ν−22\nu-2 essential constants, and quotienting by the three-dimensional group of projective linear changes of the variables leaves 2ν−52\nu-5, exactly the number of coefficient conditions, so a finite list of solutions is expected at each order where the count does not degenerate. Hilbert solves the cases ν=3,…,8\nu=3,\dots,8 completely: unique pencils for ν=3,4,5\nu=3,4,5; a two-parameter family for the exceptional order ν=6\nu=6, where the count degenerates; exactly two inequivalent pencils for ν=7\nu=7; and exactly three for ν=8\nu=8. The Jacobians of the unique pencils of orders ν=3,4,5\nu=3,4,5 are the classical binary forms attached to the finite rotation groups of the tetrahedron, octahedron and hexahedron (of orders 4, 6 and 8), whose fourth self-transvectants vanish or reproduce the form, and the Jacobian of one of the two order-7 pencils is the icosahedron form of order 12; at the exceptional order ν=6\nu=6 the Jacobian is a form of order 10 depending on the two family parameters, and at ν=8\nu=8 the Jacobian of one of the three pencils is a product of the octahedron and hexahedron forms. For arbitrary order ν\nu, Hilbert exhibits two further pencils, with generators supported on the monomials x1νx_1^\nu, x1ν−1x2x_1^{\nu-1}x_2, x1ν−2x22x_1^{\nu-2}x_2^2 and x2νx_2^\nu with ν\nu-dependent coefficients (the explicit normal forms are on p. 569 of the cited paper), but he remarks that with these the enumeration of all pencils with identically vanishing third transvectant is probably not yet closed. The same paper poses, for even-order forms, the companion question of whether the Jacobians of these pencils are the only binary forms whose fourth self-transvectant (f,f)4(f,f)_4 is divisible by ff itself. A contemporary survey placing these questions in the invariant theory of the period is F. Meyer's report in the Jahresbericht of the German Mathematical Society, which also records the early history of the divisibility question.

Problem Statement

Determine, for every order ν≥3\nu\ge 3, all pencils {λφ+μψ}\{\lambda\varphi+\mu\psi\} of binary forms of order ν\nu over the complex numbers whose third transvectant (φ,ψ)3(\varphi,\psi)_3 vanishes identically, up to equivalence under linear change of the variables and change of basis of the pencil, excluding the trivial pencils with degenerate Jacobian covariant: namely, those whose forms all share one or more common linear factors, and those in which forms with linear factors of multiplicity greater than two occur (Hilbert 1887, Math. Ann. 30, p. 563). For ν≤8\nu\le 8 the answer is known (Hilbert 1887): unique pencils for ν=3,4,5\nu=3,4,5, a two-parameter family for the exceptional order ν=6\nu=6, two inequivalent pencils for ν=7\nu=7, and three for ν=8\nu=8; two explicit families exist for every order ν\nu, but Hilbert states that with them the enumeration is probably not closed. The open question is whether, at every order ν\nu, all such pencils are exhausted by Hilbert's two explicit families together with the low-order exceptions, or whether additional pencils occur at some orders and, if so, what they are. The companion divisibility question from the same paper — whether the Jacobians of these pencils are the only even-order forms whose fourth self-transvectant is divisible by the form itself — has been answered affirmatively (A. Hirsch, Königsberg dissertation 1892, as attested by Hilbert's footnote and by F. Meyer's report) and is not part of what remains open.

References

  1. D. Hilbert, "Ueber binäre Formenbüschel mit besonderer Combinanteneigenschaft", Mathematische Annalen 30 (1887), 561–570. DOI 10.1007/bf01444097, https://doi.org/10.1007/bf01444097 (digitized copy: https://gdz.sub.uni-goettingen.de/fulltext/PPN235181684_0030/00000587.xml)
  2. D. Hilbert, "Ueber Büschel von binären Formen mit vorgeschriebener Functionaldeterminante", Mathematische Annalen 33 (1888), 227–236. DOI 10.1007/bf01443853, https://doi.org/10.1007/bf01443853
  3. E. Stroh, "Ueber eine fundamentale Eigenschaft des Ueberschiebungs-processes und deren Verwerthung in der Theorie der binären Formen", Mathematische Annalen 33 (1888), 61–107. DOI 10.1007/bf01444111, https://doi.org/10.1007/bf01444111
  4. Franz Meyer, "Bericht über den gegenwärtigen Stand der Invariantentheorie", Jahresbericht der Deutschen Mathematiker-Vereinigung, Band 1 (1890–1892). https://resolver.sub.uni-goettingen.de/purl?PPN37721857X_0001 (digitized copy: https://gdz.sub.uni-goettingen.de/fulltext/PPN37721857X_0001/00000087.xml)
  5. A. Rocco, "Ricerche sulle forme binarie aventi zero le 6e, le 8e, o le 2ime spinte su sè stesse", Giornale di Matematiche di Battaglini, III. Serie, 61 (1923), 129–148. zbMATH/JFM identifier JFM 49.0695.01, https://zbmath.org/?q=an%3A49.0695.01
  6. G. Fano, "Sulle forme binarie per le quali una delle spinte in sè stesse sia identicamenta nulla", Giornale di Matematiche di Battaglini, III. Serie, 62 (1924), 91–98. zbMATH/JFM identifier JFM 50.0056.01, https://zbmath.org/?q=an%3A50.0056.01
  7. Abdelmalek Abdesselam and Jaydeep Chipalkatti, "On the Linear Combinants of a Binary Pencil", arXiv:0802.3149 (2008), https://arxiv.org/abs/0802.3149
  8. Abdelmalek Abdesselam and Jaydeep Chipalkatti, "The Higher Transvectants are Redundant", arXiv:0801.1533 (2008), https://arxiv.org/abs/0801.1533
  9. Abdelmalek Abdesselam and Jaydeep Chipalkatti, "Quadratic Involutions on Binary Forms", arXiv:1008.3117 (2010), https://arxiv.org/abs/1008.3117
  10. Abdelmalek Abdesselam and Jaydeep Chipalkatti, "On Hilbert Covariants", arXiv:1203.4761 (2012), https://arxiv.org/abs/1203.4761