Hilbert's 1894 problem on the minimal $L^2$ norm of an integer polynomial: the critical interval length $\ge 4$

For a real interval $[a,b]$, let $m(a,b) = \inf_{0 \ne P \in \mathbb{Z}[x]} \int_a^b P(x)^2\,dx$, where $\mathbb{Z}[x]$ is the set of polynomials with integer coefficients. In an 1894 paper that also introduced the Hilbert matrix, David Hilbert asked whether this infimum can be made smaller than any prescribed $\varepsilon > 0$, and proved that it equals $0$ whenever $b - a < 4$: the asymptotics of the Gram determinant of the monomial basis (computed through Legendre polynomials), combined with Minkowski's theorem on lattice points in convex bodies, yield integer polynomials whose squared $L^2$ norm decays exponentially. The complementary case — intervals of length $\ge 4$, whose logarithmic capacity $(b-a)/4$ is $\ge 1$ — has remained unresolved since 1894: neither a proof that the infimum stays $0$, nor a positive lower bound, is known. The cited literature records active adjacent research lines — the sup-norm integer Chebyshev problem, the distribution of conjugate algebraic integers, and the arithmetic of inverse Gram matrices arising in control theory — but no progress on the critical case itself, no general constructive methods, and no exact fixed-degree minima.

Background

The problem sits at the interface of Diophantine approximation (how well objects defined by integrality constraints can approximate analytic ones) and the analytic theory of polynomials with integer coefficients. A polynomial in $\mathbb{Z}[x]$ has integer coefficients and is called an integer (or integral) polynomial. For a closed interval $[a,b]$, the $L^2$ norm of a polynomial $P$ is $\|P\|_{L^2[a,b]} = \left(\int_a^b P(x)^2\,dx\right)^{1/2}$; the object of study is the squared norm $\int_a^b P(x)^2\,dx$, a positive real number for every nonzero $P$.

Two classical structures enter. First, the Gram matrix of the monomials $1, x, \ldots, x^{N-1}$ in $L^2[a,b]$, with entries $\int_a^b x^{i+j}\,dx$; on $[0,1]$ this is the Hilbert matrix $(1/(i+j+1))_{i,j \ge 0}$, which Hilbert introduced in the same 1894 paper. Second, the coefficient vector $v \in \mathbb{Z}^{N}$ of a polynomial of degree $< N$ determines $\int_a^b P^2\,dx$ as a positive definite quadratic form $Q(v)$ in the coefficients; finding the minimum at fixed degree is a shortest-vector problem for a lattice (finding the nonzero lattice vector on which a positive definite quadratic form takes its least value). Hilbert evaluated the determinant of this quadratic form exactly (via the Legendre polynomial basis — the standard orthogonal polynomials on $[-1,1]$) and showed that its asymptotics cross the applicability threshold of Minkowski's theorem on lattice points in centrally symmetric convex bodies (the fact that such a body in $\mathbb{R}^N$ whose volume exceeds $2^N$ times the covolume of the lattice contains a nonzero lattice point) exactly when the interval length $b - a$ equals $4$, concluding $m(a,b) = 0$ for $b - a < 4$ (as recounted by Pritsker 2005 and by Ovseevich 2021, who quotes Hilbert's question verbatim).

The $L^\infty$ analogue is the integer Chebyshev problem: $t_\mathbb{Z}([a,b]) = \lim_{n \to \infty}\left(\min_{0 \ne P \in \mathbb{Z}[x],\ \deg P \le n}\max_{x \in [a,b]}|P(x)|\right)^{1/n}$. Because the $L^2$ and $L^\infty$ norms of degree-$n$ polynomials on an interval differ only by a subexponential-in-$n$ factor, Hilbert's $L^2$ result implies the Hilbert–Fekete bound $t_\mathbb{Z}([a,b]) \le \sqrt{(b-a)/4}$. The quantity $(b-a)/4$ is the logarithmic capacity $\mathrm{cap}([a,b])$ of the interval (a conformally invariant measure of its size), and the dichotomy $\mathrm{cap} < 1$ versus $\mathrm{cap} \ge 1$ — equivalently interval length below versus at least $4$ — recurs throughout this circle of problems: for the sup norm one knows $t_\mathbb{Z}([a,b]) = 1$ exactly when $b - a \ge 4$, while for $b - a < 4$ the exact value is unknown for every segment (even on $[0,1]$ only $0.4213 < t_\mathbb{Z}([0,1]) < 0.42291334$ is known; Pritsker 2012), and the same threshold governs how many complete sets of conjugate algebraic integers (the roots of a single irreducible monic polynomial with integer coefficients form one such set) can lie in an interval (work of Schur, P'olya, Fekete, Robinson; cf. Orloski–Sardari 2023). The $L^2$ question at the threshold itself, however, is untouched by all of these results.

Problem Statement

Let $\mathbb{Z}[x]$ denote the polynomials in one real variable with integer coefficients. For a closed interval $[a,b] \subset \mathbb{R}$ and an integer $n \ge 0$, define

$$m(a,b) = \inf_{0 \ne P \in \mathbb{Z}[x]} \int_a^b P(x)^2\,dx, \qquad m_n(a,b) = \min_{0 \ne P \in \mathbb{Z}[x],\ \deg P \le n} \int_a^b P(x)^2\,dx.$$

The minimum $m_n$ is attained, since the values $\int_a^b P^2\,dx$ are the values of a positive definite quadratic form on the integer lattice of coefficient vectors, a discrete set. Hilbert (1894) proved $m(a,b) = 0$ whenever $b - a < 4$; the infimum question for length $\ge 4$ is the open problem transmitted by the sources cited below, and two adjacent quantitative aspects of the length $< 4$ regime are listed after it for context:

(1) Critical case. Decide whether $m(a,b) = 0$ or $m(a,b) > 0$ for intervals with $b - a \ge 4$. Note that $m$ is monotone under interval inclusion — $[a,b] \subseteq [c,d]$ implies $m(a,b) \le m(c,d)$, since $\int_a^b P^2 \le \int_c^d P^2$ for each $P$ — so asserting a positive lower bound for all intervals of length $\ge 4$ is equivalent to asserting it for all intervals of length exactly $4$ (every longer interval contains one). The converse direction does not reduce: $m = 0$ on a given length-4 interval does not by itself imply $m = 0$ on superintervals. Also, $m(a+k, b+k) = m(a,b)$ for integer shifts $k$, but a general translation does not preserve $\mathbb{Z}[x]$, so the answer may depend on the position of the interval, not only its length; the required decision covers arbitrary $[a,b]$ with $b - a \ge 4$, including rational and irrational endpoints.

(2) Constructive aspect. Hilbert's argument establishes existence non-constructively through Minkowski's theorem on lattice points in centrally symmetric convex bodies. This aspect concerns explicit sequences of polynomials $P_n \in \mathbb{Z}[x]$, or a deterministic algorithm producing them, such that $\int_a^b P_n^2\,dx \to 0$ for intervals of length below $4$, together with the dependence of the construction on the endpoints $a, b$. Trivial explicit families such as the shifted Legendre polynomials $\mathrm{P}_n(2x-1) \in \mathbb{Z}[x]$ (with $\mathrm{P}_n$ the $n$-th Legendre polynomial), which satisfy $\int_0^1 \mathrm{P}_n(2x-1)^2\,dx = 1/(2n+1) \to 0$ on $[0,1]$, decay only polynomially in $n$ and only on special intervals; constructions achieving the exponential decay rate $((b-a)/4 + o(1))^n$ that Hilbert's method predicts, for general endpoints, belong to this aspect.

(3) Quantitative aspect. The exact minimum $m_n(a,b)$ at fixed degree $n$ (a shortest-vector problem for the Gram matrix of $\{1, x, \ldots, x^n\}$ in $L^2[a,b]$: finding the nonzero integer coefficient vector on which the positive definite quadratic form $\int_a^b P(x)^2\,dx$ takes its least value), asymptotic constants for $m_n(a,b)^{1/n}$, their dependence on the endpoints, and their behavior at the critical length $b - a = 4$ belong to this aspect.

The problem is a resolution of item (1): a decision, for intervals $[a,b]$ with $b - a \ge 4$, of whether $m(a,b) = 0$ or $m(a,b) > 0$. Since general translations do not preserve $\mathbb{Z}[x]$, the answer may depend on the position of the interval and not only on its length, so intervals of length $\ge 4$ need not behave alike. Items (2) and (3) are adjacent quantitative aspects of the same circle of problems — understanding how small the $L^2$ norm of a nonzero integer polynomial on $[a,b]$ can be — retained here as background: they are not required components of a resolution of item (1).

References

1. D. Hilbert, “Ein Beitrag zur Theorie des Legendre'schen Polynoms”, Acta Mathematica 18 (1894), 155–159. DOI: 10.1007/BF02418278, https://doi.org/10.1007/BF02418278.
2. I. Schur, “Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten”, Mathematische Zeitschrift 1 (1918), 377–402. DOI: 10.1007/BF01465096, https://doi.org/10.1007/BF01465096 (with Pólya, the classical source of the finiteness theorem for complete sets of conjugate algebraic integers in intervals of length below 4).
3. M. Fekete, “Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten”, Mathematische Zeitschrift 17 (1923), 228–249. DOI: 10.1007/BF01504345, https://doi.org/10.1007/BF01504345 (transfinite-diameter generalization to arbitrary compact sets).
4. G. Szegő, “Bemerkungen zu einer Arbeit von Herrn M. Fekete: Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten”, Mathematische Zeitschrift 21 (1924), 203–208. DOI: 10.1007/BF01187465, https://doi.org/10.1007/BF01187465 (equality of transfinite diameter and logarithmic capacity).
5. R. M. Robinson, “Conjugate algebraic integers in real point sets”, Mathematische Zeitschrift 84 (1964), 415–427. DOI: 10.1007/BF01109909, https://doi.org/10.1007/BF01109909 (intervals of length greater than 4 carry infinitely many complete sets of conjugates).
6. F. Amoroso, “Sur le diamètre transfini entier d'un intervalle réel”, Annales de l'Institut Fourier 40 (1990), 885–911. DOI: 10.5802/aif.1240, https://doi.org/10.5802/aif.1240 (refinement of Hilbert's approach for intervals with rational endpoints).
7. B. S. Kashin, “Algebraic polynomials with integer coefficients deviating little from zero on an interval”, Mathematical Notes of the Academy of Sciences of the USSR 50 (1991), 921–927. DOI: 10.1007/BF01156135, https://doi.org/10.1007/BF01156135 (refinement for symmetric intervals).
8. I. E. Pritsker, “Small polynomials with integer coefficients”, Journal d'Analyse Mathématique 96 (2005), 151–190. DOI: 10.1007/BF02787827; arXiv:math/0101166, https://arxiv.org/abs/math/0101166 (contains the historical account of Hilbert's 1894 result and its sup-norm consequences).
9. I. E. Pritsker, “Polynomials with integer coefficients and their zeros”, Journal of Mathematical Sciences 183 (2012), 810–822. DOI: 10.1007/s10958-012-0842-z; arXiv:1307.6200, https://arxiv.org/abs/1307.6200 (survey of the integer Chebyshev problem and the state of its exact values).
10. P. B. Borwein, C. G. Pinner, I. E. Pritsker, “Monic integer Chebyshev problem”, Mathematics of Computation 72 (2003), 1901–1916. DOI: 10.1090/S0025-5718-03-01477-7; arXiv:1307.5362, https://arxiv.org/abs/1307.5362 (sup-norm monic analogue).
11. P. B. Borwein, I. E. Pritsker, “The multivariate integer Chebyshev problem”, Constructive Approximation 30 (2009), 299–310. DOI: 10.1007/s00365-008-9029-8; arXiv:1307.5456, https://arxiv.org/abs/1307.5456 (multivariate sup-norm generalization of the Hilbert–Fekete bound).
12. K. G. Hare, P. W. Hodges, “Applications of Integer and Semi-Infinite Programming to the Integer Chebyshev Problem” (arXiv preprint title; the version of record is listed as “Applications of Integer Semi-Infinite Programing to the Integer Chebyshev Problem”), Experimental Mathematics 31 (2022), no. 2, 694–700. DOI: 10.1080/10586458.2019.1691089, https://doi.org/10.1080/10586458.2019.1691089; arXiv:1804.05985, https://arxiv.org/abs/1804.05985 (computational state of the art for sup-norm integer Chebyshev polynomials on $[0,1]$).
13. A. Fedorov, A. Ovseevich, “Asymptotic Control Theory for a System of Linear Oscillators”, Moscow Mathematical Journal 16 (2016), no. 3, 561–598. DOI: 10.17323/1609-4514-2016-16-3-561-598; arXiv:1308.6090, https://arxiv.org/abs/1308.6090 (credits the inverse-Gram-matrix idea to Hilbert's 1894 paper).
14. A. I. Ovseevich, “Control Theory, Integral Matrices, and Orthogonal Polynomials” (Теория управления, целочисленные матрицы и ортогональные полиномы), Proceedings of the Steklov Institute of Mathematics 315 (2021), no. 1, 161–170 (English translation); Russian original: Trudy Matematicheskogo Instituta imeni V.A. Steklova 315 (2021), 172–181. DOI (English): 10.1134/S0081543821050126, https://doi.org/10.1134/S0081543821050126; DOI (Russian): 10.4213/tm4224, https://doi.org/10.4213/tm4224 (quotes Hilbert's question verbatim and names “the Hilbert problem on the minimal $L^2$ norm of an integral polynomial” as motivation).
15. A. I. Ovseevich, “A number-theoretic part of control theory”, Russian Mathematical Surveys 77:2 (2022), 369–371. DOI: 10.1070/RM10050, https://doi.org/10.1070/RM10050 (divisibility theorems for inverse Gram matrices via shifted Jacobi polynomials and the Christoffel–Darboux kernel).
16. B. J. Orloski, N. T. Sardari, “Limiting distributions of conjugate algebraic integers”, arXiv:2302.02872, https://arxiv.org/abs/2302.02872 (representative of the adjacent conjugate-algebraic-integers line sharing the capacity-1 threshold).