Carl Ludwig Siegel Quotes

I am afraid that mathematics will perish by the end of this century if the present trend for senseless abstraction — as I call it: theory of the empty set — cannot be blocked up.

Ein Bourgeois, wer noch Algebra treibt! Es lebe die unbeschrankte Individualitat der transzendenten Zahlen! ["It's a bourgeois, who still does algebra! Long live the unrestricted individuality of transcendental numbers!"]

One of the many importants ideas introduced by Minkowski into the study of convex bodies was that of gauge function. Roughly, the gauge function is the equation of a convex body. Minkowski showed that the gauge function could be defined in a purely geometric way and that it must have certain properties analogous to those possessed by the distance of a point from the origin. He also showed that conversely given any function possessing these properties, there exists a convex body with the given function as its gauge function.

Ours, according to Leibnitz, is the best of all possible worlds, and the laws of nature can therefore be described in terms of extremal principles. Thus, arising from corresponding variational problems, the differential equations of mechanics have invariance properties relative to certain groups of coordinate transformations.

Let θ be an algebraic integer and assume that all conjugates of θ, except θ itself, have an absolute value less than 1. Then –θ also has this property; on the other hand, θ is real. Without loss of generality, we may therefore suppose θ ≥ 0. Since the norm of θ is a rational integer, we have θ ≥ 1, except for the trivial case θ = 0. Recently, R. Salem ... discovered the interesting theorem that the set S of all θ is closed and that θ = 1 is an isolated point of S. Consequently there exists a smallest θ = θ₁ > 1. We shall prove that θ₁ is the positive zero of x³ – x – 1 and that also θ₁ is isolated in S. Moreover we shall prove that the next number of S, namely the smallest θ = θ₂ > θ₁, is the positive zero of x⁴ – x³ – 1 and that θ₂ is again an isolated point of S. Since θ₁ = 1.324..., θ₂ = 1.380..., both numbers are less than 2^½; therefore our statements are contained in the following: . Let θ be an algebraic integer whose conjugates lie in the interior of the unit circle; if ±θ ≠ 0, 1, θ₁, θ₂, then θ² > 2.

The theory of functions of several variables turns out to be essentially more difficult than the theory of one variable because of the existence of points of indeterminacy. In the case n > 1, a mere glance at the poles already indicates a behavior which is completely different from that in the case n = 1. The reason is that, in case n > 1, the poles are not isolated and, in general, there does not exist a Laurent expansion. In a neighborhood of a nonregular point we are forced to view meromorphic functions as quotients of power series.