In an 1888 paper in Mathematische Annalen, David Hilbert gave a uniform algebraic solution, valid for every order vv, of the classical problem of determining all pencils (two-dimensional linear families) of binary forms of degree vv whose functional determinant — the Jacobian of a basis pair, a binary form of degree 2v−22v-2 — coincides with a given form ff; equivalently, all involutions of order vv on the projective line whose 2v−22v-2 double elements are prescribed. His elimination produces a characteristic equation F(λ)=0F(\lambda)=0 of degree (2v−2)!/((v−1)! v!)(2v-2)!/((v-1)!\,v!) (a Catalan number) whose roots parametrize the sought pencils. Hilbert states explicitly that complete settlement requires investigating all degenerations (Ausartungen) caused by the coincidence of several roots of this equation, and he treats only the case of a single double root, distinguishing two subcases: the condition J=0\mathfrak{J}=0 makes λ=0\lambda=0 a double root and two pencils genuinely coincide, versus a root value that additionally annihilates an auxiliary expression of the construction, in which case the double root carries two distinct pencils recoverable by solving a quadratic equation. The problem's modern descendant — the Wronski map from the Grassmannian of two-planes in SymvC2\mathrm{Sym}^v\mathbb{C}^2 to the space of binary forms of degree 2v−22v-2 — has known generic degree (the Catalan number, by Eremenko–Gabrielov), known solution counts at generic positions with multiple critical points (Kostka numbers, by Scherbak), and known reality theorems (Eremenko–Gabrielov and others), but no classification of its branch locus or degenerate fibers is known, classically or in the modern literature. Completing Hilbert's degeneration analysis for all root-collision patterns of F(λ)=0F(\lambda)=0 and all orders vv remains open.

Background

A binary form of degree vv (order vv in the classical language) is a homogeneous polynomial φ(x,y)\varphi(x,y) of degree vv in two variables. A pencil (German: Büschel) of such forms is the set {λφ+μψ}\{\lambda\varphi+\mu\psi\} of all linear combinations of two linearly independent forms, considered up to a change of basis; geometrically it is a line in the projective space of degree-vv binary forms. The functional determinant of the pencil is the Jacobian J(φ,ψ)=∂(φ,ψ)/∂(x,y)J(\varphi,\psi)=\partial(\varphi,\psi)/\partial(x,y), a binary form of degree 2v−22v-2 that is independent of the choice of basis up to a constant factor; in modern terms it is the Wronskian of the two-dimensional subspace spanned by φ,ψ\varphi,\psi. A covariant is a form built equivariantly from a given form's coefficients and variables; a combinant is a joint covariant of a pair of forms that depends only on the pencil they span (up to a power of the change-of-basis determinant), so combinants are the natural invariants of a pencil. An involution of order vv on the projective line is a one-dimensional linear system of divisors of degree vv; equivalently, a rational function of degree vv on the Riemann sphere considered up to post-composition with a fractional linear (Möbius) transformation. The pencil is the associated linear system of forms, and the zeros of the Jacobian are the 2v−22v-2 branch points (classically, the double elements) of the corresponding map.

The classical problem, posed and treated for low orders by Stéphanos (1881; the quartic case was also treated by Brill, 1882), asks: given a binary form ff of degree 2v−22v-2, find all pencils of degree-vv forms with functional determinant ff. The number of such pencils for generic ff had been obtained just before by three independent classical computations (recalled in the introduction of Hilbert's 1888 paper) as (2v−2)!/((v−1)! v!)(2v-2)!/((v-1)!\,v!) — the Catalan number Cv−1C_{v-1} (for v=3v=3 there are two pencils; for v=4v=4, five).

Hilbert's 1888 paper gives a uniform algebraic solution for every order vv. A form ff of degree 2v−22v-2 being given, he constructs a simultaneous invariant S(f,φ,ψ)S(f,\varphi,\psi) of three forms (of degree v−2v-2 in each argument) with the property that, upon substituting a basis pair φ,ψ\varphi,\psi of a pencil, the identity forces the vanishing of a determinant that is linear in the combinant λ=J(φ,ψ)\lambda=\mathfrak{J}(\varphi,\psi) (Stéphanos's combinant, of degree v−2v-2 in each form's coefficients and distinct from the Jacobian J(φ,ψ)J(\varphi,\psi); the condition J=0\mathfrak{J}=0 characterizes pairs whose pencils can share a Jacobian). Eliminating the basis-dependent variables produces a determinant equation of the shape D(λ)=F(λ)v−1=0D(\lambda)=F(\lambda)^{v-1}=0, and the characteristic equation F(λ)=0F(\lambda)=0 — of degree Cv−1C_{v-1} with coefficients that are integral invariants of ff — has as its roots the parameter values λ\lambda of the sought pencils, each root determining one pencil. For v=4v=4 his equation turns into Stéphanos's quintic by a simple substitution.

For generic ff the roots of FF are distinct and the Catalan count is recovered. When ff is special, roots of FF can coincide, and Hilbert's closing paragraph states that complete settlement requires investigating all such degenerations (Ausartungen), while he analyses only a single double root, with two subcases: (i) the condition J=0\mathfrak{J}=0 makes the value λ=0\lambda=0 a double root, and two of the sought pencils genuinely coincide; (ii) a root value λ\lambda simultaneously annihilates an auxiliary expression of the construction (a determinant bordered by the coefficients of ff), and the double root then carries two distinct pencils, constructed by solving a quadratic equation. All other collision patterns — several double roots at once, triple and higher multiplicities, and their combinations — are left untreated, and no strata in the space of forms ff carrying each degeneration type are identified.

In modern language the problem is the study of the Wronski map Wr:Gr(2,SymvC2)→P(Sym2v−2C2)\mathrm{Wr}:\mathrm{Gr}(2,\mathrm{Sym}^v\mathbb{C}^2)\to\mathbb{P}(\mathrm{Sym}^{2v-2}\mathbb{C}^2), which sends a two-dimensional subspace of degree-vv binary forms (a point of the Grassmannian, the projective variety parametrizing such subspaces) to its Wronskian determinant. The fiber over [f][f] is exactly the set of pencils with functional determinant ff, up to the natural equivalence. The map is finite of degree Cv−1C_{v-1}; its complex degree was computed by Eremenko–Gabrielov (2002), confirming the classical count uniformly, and its real degrees are also known (Goldberg's branched-covering count is related). Finiteness implies the map is flat, so every scheme-theoretic fiber has total length Cv−1C_{v-1} when counted with multiplicity — the degeneration question is thus precisely the description of the support and nilpotent structure of the fibers over special forms, and of the branch locus in the target. The modern literature has determined: the number of classes of degree-vv rational functions with prescribed critical points at generic positions with prescribed multiplicities (Scherbak 2002: a Kostka number, i.e., a multiplicity in a tensor product of sl2\mathfrak{sl}_2-representations, reducing to Cv−1C_{v-1} when all critical points are simple); reality of the solutions for real data (the B. and M. Shapiro conjecture for rational functions, proved by Eremenko–Gabrielov 2002, with a topological proof by Levinson–Purbhoo 2021); existence in degenerate real configurations via nets of rational functions (Eremenko–Gabrielov–Shapiro–Vainshtein); and the representation-theoretic theory of pencil combinants, including the reconstruction of a pencil from its first and third linear combinants (Abdesselam–Chipalkatti). Crooks–Milson studied the equivariance of the Wronski map for the equivalence problem of polynomial subspaces. None of these works describes the degenerate fibers or the branch locus of the Wronski map, which is precisely Hilbert's untreated Ausartungen program in modern form.

Problem Statement

Hilbert (1888) leaves open the complete investigation of the degenerations (Ausartungen) of his solution to the prescribed-functional-determinant problem — those caused by the coincidence of several roots of the characteristic equation F(λ)=0F(\lambda)=0. The open problem, for every order v≥2v\ge 2 and every pattern of root coincidence beyond the single double root (several simultaneous double roots, triple and higher multiplicities, and their combinations), is to: (1) determine which of the sought pencils genuinely coincide (the analogue of Hilbert's subcase where J=0\mathfrak{J}=0 makes λ=0\lambda=0 a double root) and which remain distinct while the root-to-pencil parametrization degenerates (the analogue of his subcase where a root value also annihilates the auxiliary expression, two distinct pencils being attached to a double root and recoverable from a quadratic), including the equations (resolvents of appropriate degree) that reconstruct the distinct pencils attached to a multiple root; (2) identify the loci in the coefficient space of the given form ff of degree 2v−22v-2 on which each degeneration type occurs (the discriminant-type stratification of the space of Wronskians); and (3) describe the limiting structure of the full set of pencils for arbitrary specializations of ff, including the scheme-theoretic (nilpotent) structure of non-reduced fibers, whose total length must remain Cv−1=(2v−2)!/((v−1)! v!)C_{v-1}=(2v-2)!/((v-1)!\,v!) by flatness. In modern terms this is the classification of the branch locus, ramification, and degenerate fibers of the Wronski map Wr:Gr(2,SymvC2)→P(Sym2v−2C2)\mathrm{Wr}:\mathrm{Gr}(2,\mathrm{Sym}^v\mathbb{C}^2)\to\mathbb{P}(\mathrm{Sym}^{2v-2}\mathbb{C}^2) for general vv, together with its translation back into Hilbert's characteristic-equation parametrization. The classification is sought for every order v≥2v\ge 2, the cases v=3v=3 and v=4v=4 (where FF is quadratic and quintic respectively) being instances of it rather than the full target, and it recovers Hilbert's single-double-root analysis (both subcases) and the generic Catalan count as special cases.

References

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