Scientists / Open questions

David Hilbert

Number theory; foundations of geometry; mathematical logic

13 questions

Mutual arrangement of the ovals of nonsingular real plane algebraic curves: classification of the realizable real schemes (the plane-curve part of part I of Hilbert's sixteenth problem)
Hilbert's 1894 problem on the minimal $L^2$ norm of an integer polynomial: the critical interval length $\ge 4$
Convergence and all-time validity of Hilbert's power-series solution of the Boltzmann equation: completing Hilbert's 1912 integral-equation foundation of the kinetic theory of gases
Hilbert's classes of differential equations: omitted solutions, discriminating solutions, and the classification of underdetermined equations into classes
Effective and quantitative forms of Hilbert's irreducibility theorem: polynomial specialization bounds and complete arithmetic description of exceptional sets
Effective full invariant systems and sharp weight bounds for ternary forms of general order
Explicit classical-covariant expansion of Hilbert's power-detection covariants $C_\nu$ for binary forms, for arbitrary $\nu$
Hilbert's 1902 motion-based foundations of plane geometry: the open questions of axiom redundancy, the spatial analogue, elliptic geometry, and the replacement of Axiom III
Hilbert's twofold overshift (the case $\kappa=2$ of the 1887 invariant-theoretic program): the eigenform theory of binary forms of odd order, and the deferred most general degeneration of the $\kappa=1$ theory
Complete classification of binary form pencils with identically vanishing third transvectant for general order (Hilbert 1887)
Extension of Hilbert's inner-uniqueness sufficiency criterion to variational problems with non-prescribed boundary values
Pencils of binary forms with prescribed discriminant: constructive determination and enumeration for arbitrary degree
Complete classification of the degenerations (Ausartungen) of Hilbert's problem on pencils of binary forms with a prescribed functional determinant

Back to open problems

Mutual arrangement of the ovals of nonsingular real plane algebraic curves: classification of the realizable real schemes (the plane-curve part of part I of Hilbert's sixteenth problem)

Hilbert's 1894 problem on the minimal L2L^2L2 norm of an integer polynomial: the critical interval length ≥4\ge 4≥4

Convergence and all-time validity of Hilbert's power-series solution of the Boltzmann equation: completing Hilbert's 1912 integral-equation foundation of the kinetic theory of gases

Hilbert's classes of differential equations: omitted solutions, discriminating solutions, and the classification of underdetermined equations into classes

Effective and quantitative forms of Hilbert's irreducibility theorem: polynomial specialization bounds and complete arithmetic description of exceptional sets

Effective full invariant systems and sharp weight bounds for ternary forms of general order

Explicit classical-covariant expansion of Hilbert's power-detection covariants CνC_\nuCν​ for binary forms, for arbitrary ν\nuν

Hilbert's 1902 motion-based foundations of plane geometry: the open questions of axiom redundancy, the spatial analogue, elliptic geometry, and the replacement of Axiom III

Hilbert's twofold overshift (the case κ=2\kappa=2κ=2 of the 1887 invariant-theoretic program): the eigenform theory of binary forms of odd order, and the deferred most general degeneration of the κ=1\kappa=1κ=1 theory

Complete classification of binary form pencils with identically vanishing third transvectant for general order (Hilbert 1887)

Extension of Hilbert's inner-uniqueness sufficiency criterion to variational problems with non-prescribed boundary values

Pencils of binary forms with prescribed discriminant: constructive determination and enumeration for arbitrary degree

Complete classification of the degenerations (Ausartungen) of Hilbert's problem on pencils of binary forms with a prescribed functional determinant

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

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

Hilbert's twofold overshift (the case $\kappa=2$ of the 1887 invariant-theoretic program): the eigenform theory of binary forms of odd order, and the deferred most general degeneration of the $\kappa=1$ theory