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Mathematical Problems

Chapter 18: 17. EXPRESSION OF DEFINITE FORMS BY SQUARES.
By David Hilbert
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About This Book

A systematic collection of fundamental open problems is presented to chart directions for future mathematical research. Topics range across set theory, the consistency of arithmetic, geometry, transformation groups, number theory, algebraic forms, analysis, the calculus of variations, and the mathematical formulation of physical axioms. Challenges highlighted include questions about the continuum, irrationality and transcendence, the distribution and properties of primes, reciprocity laws, solvability of Diophantine equations, uniformization and boundary-value problems, and the finiteness or structure of function systems. Each problem is posed to emphasize its technical difficulty and its capacity to stimulate new methods and broader theoretical development.

17. EXPRESSION OF DEFINITE FORMS BY SQUARES.

A rational integral function or form in any number of variables with real coefficients such that it becomes negative for no real values of these variables, is said to be definite. The system of all definite forms is invariant with respect to the operations of addition and multiplication, but the quotient of two definite forms—in case it should be an integral function of the variables—is also a definite form. The square of any form is evidently always a definite form. But since, as I have shown,[38] not every definite form can be compounded by addition from squares of forms, the question arises—which I have answered affirmatively for ternary forms[39]—whether every definite form may not be expressed as a quotient of sums of squares of forms. At the same time it is desirable, for certain questions as to the possibility of certain geometrical constructions, to know whether the coefficients of the forms to be used in the expression may always be taken from the realm of rationality given by the coefficients of the form represented.[40]

I mention one more geometrical problem:

[38] Math. Annalen, vol. 32.

[39] Acta Mathematica, vol. 17.

[40] Cf. Hilbert: Grunglagen der Geometrie, Leipzig, 1899, Chap. 7 and in particular § 38.