Fix an integer ν ≥ 2 and a binary form δ(λ,μ) of degree 2ν−2, i.e. a homogeneous polynomial in two variables. The problem is to determine all pencils λφ+μψ of binary ν-ic forms φ,ψ — one-parameter linear families — whose discriminant, viewed as a binary form of degree 2ν−2 in the pencil parameters (λ,μ), equals the prescribed form δ, where pencils differing only by a nonsingular linear change of the underlying variables are identified. Hilbert (1888) proved indirectly that every δ arises this way and from only finitely many pencils, and solved the constructive problem for ν = 2 (exactly one pencil), ν = 3 (four pencils when the prescribed form has distinct roots, via a tetrahedral equation and covariants of a quartic), and ν = 4 (120 pencils when the prescribed form has distinct roots, via the Cayley–Clebsch decomposition of a binary sextic and Hessians of ternary cubics); the degenerate cases, where δ has multiple linear factors, he treated only in passing. For arbitrary ν ≥ 5 no constructive method is known, the exact number of pencils realizing a given δ is known in principle only in the squarefree case, where it reduces to a classical Hurwitz number, and no complete treatment of degenerate δ is known. Via the dictionary between pencils and rational maps, the problem is equivalent to constructing and counting rational functions of degree ν with prescribed branch divisor: the squarefree case is governed by classical Hurwitz theory, while the degenerate case contains the still-open Hurwitz existence problem.

Background

A binary form of degree ν (a binary ν-ic) is a homogeneous polynomial φ(x,y) of degree ν in two variables; its discriminant is a polynomial in the coefficients that vanishes exactly when φ acquires a multiple linear factor. A pencil of binary ν-ic forms is the one-parameter family λφ+μψ spanned by an unordered pair (φ,ψ); a member has a double root precisely when the parameter pair (λ,μ) is a root of the discriminant form δ(λ,μ), a binary form of degree 2ν−2 in (λ,μ). Hilbert's problem prescribes this form and asks for all pencils realizing it, up to linear changes of the variables (x,y) — a three-dimensional symmetry, namely the projective linear group PGL₂(ℂ) of invertible 2×2 matrices modulo nonzero scalar matrices, which leaves every discriminant form invariant up to a fixed factor. Because a pair (φ,ψ) with no common factor defines a rational map f = φ/ψ of degree ν from the projective line to itself, the problem is equivalent to prescribing the branch divisor of a degree-ν map: the multiplicity of a root of δ equals the Riemann–Hurwitz weight Σ(e_p−1) of the corresponding critical value, where e_p are the ramification indices (local degrees) over it. Degenerate pencils, in which φ and ψ share a linear factor, correspond to degenerate covers and are one mechanism by which discriminant forms with multiple roots arise; multiple roots of δ also arise for coprime pairs whose map has Riemann–Hurwitz weight Σ(e_p−1) ≥ 2 over some value. In modern language the squarefree case (δ with 2ν−2 distinct roots) belongs to Hurwitz theory: Riemann's existence theorem gives existence, and the number of covers with fixed branch points is a Hurwitz number, computable by symmetric-group character formulas. Prescribing branch data with nontrivial ramification indices — the degenerate case — is the Hurwitz existence problem, which asks which combinatorial branch data are realizable by an actual cover; it is open in general, with new obstructions and partial solutions appearing through 2015–2026. Hilbert's problem must also not be confused with two adjacent questions: its sibling (pencils with prescribed functional determinant, i.e. Jacobian determinant ∂(φ,ψ)/∂(x,y), treated in Hilbert's 1887 note and carefully distinguished by Hilbert himself in the cited paper), and the arithmetic question of binary forms with given integer discriminant (reduction theory and invariant orders in number theory), which is adjacent but different.

Problem Statement

Let ν ≥ 2 be an integer and let δ(λ,μ) be a binary form of degree 2ν−2 over ℂ. Determine all pencils λφ+μψ of binary ν-ic forms φ,ψ — the generating pair not assumed coprime, matching Hilbert's original treatment, which explicitly allows φ and ψ to share a linear factor — such that the discriminant of the pencil member λφ+μψ, viewed as a binary form of degree 2ν−2 in the pencil parameters (λ,μ), equals the prescribed form δ(λ,μ) up to a nonzero scalar factor; two pencils are identified if they differ by a nonsingular linear change of the variables (x,y). The problem comprises: (i) a constructive method producing the pencils from the coefficients of δ; (ii) the exact number of pencils realizing each δ; and (iii) the complete analysis when δ is degenerate, i.e. has multiple linear factors (equivalently, branch data with ramification indices above 1), where the low-degree methods break down. The problem asks for all three parts (i)–(iii) for arbitrary ν and arbitrary δ, including degenerate δ. Hilbert (1888) settled (i) and (ii) for ν = 2 (one pencil), ν = 3 (four pencils) and ν = 4 (120 pencils) in the non-degenerate case where δ has distinct roots, and treated (iii) only in passing even there; he proved indirectly that existence and finiteness hold for all ν. The case ν ≥ 5 is open for all three parts, and part (iii) is open already at the level of the associated Hurwitz existence problem.

References

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