Mathematical Problems

Chapter 16: 15. RIGOROUS FOUNDATION OF SCHUBERT'S ENUMERATIVE CALCULUS.

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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.

15. RIGOROUS FOUNDATION OF SCHUBERT'S ENUMERATIVE CALCULUS.

The problem consists in this: To establish rigorously and with an exact determination of the limits of their validity those geometrical numbers which Schubert[35] especially has determined on the basis of the so-called principle of special position, or conservation of number, by means of the enumerative calculus developed by him.

Although the algebra of to-day guarantees, in principle, the possibility of carrying out the processes of elimination, yet for the proof of the theorems of enumerative geometry decidedly more is requisite, namely, the actual carrying out of the process of elimination in the case of equations of special form in such a way that the degree of the final equations and the multiplicity of their solutions may be foreseen.

[35] Kalkül der abzählenden Geometrie, Leipzig, 1879.