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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 Schubert34
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 today 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.
16.
Problem of the topology of algebraic curves and surfaces
The
maximum number of closed and separate branches which a plane algebraic
curve of the n-th order can have has been determined by Harnack.35
There arises the further question as to the relative position of the
branches in the plane. As to curves of the 6-th order, I have satisfied
myself--by a complicated process, it is true--that of the eleven branches
which they can have according to Harnack, by no means all can lie
external to one another, but that one branch must exist in whose interior
one branch and in whose exterior nine branches lie, or inversely. A
thorough investigation of the relative position of the separate branches
when their number is the maximum seems to me to be of very great interest,
and not less so the corresponding investigation as to the number, form,
and position of the sheets of an algebraic surface in space. Till now,
indeed, it is not even known what is the maxi mum number of sheets which
a surface of the 4-th order in three dimensional space can really have.36
In
connection with this purely algebraic problem, I wish to bring forward a
question which, it seems to me, may be attacked by the same method of
continuous variation of coefficients, and whose answer is of
corresponding value for the topology of families of curves defined by
differential equations. This is the question as to the maximum number and
position of Poincaré's boundary cycles (cycles limites) for a
differential equation of the first order and degree of the form
where X and Y
are rational integral functions of the n-th degree in x and
y. Written homogeneously, this is
17.
Expression of definite forms by squares
A
rational integral function or form in any number of variables with real
coefficient 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,37
not every definite form can be compounded by addition from squares of
forms, the question arises--which I have answered affirmatively for
ternary forms38--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.39 I mention one more
geometrical problem:
18.
Building up of space from congruent polyhedra
If
we enquire for those groups of motions in the plane for which a
fundamental region exists, we obtain various answers, according as the
plane considered is Riemann's (elliptic), Euclid's, or Lobachevsky's (hyperbolic).
In the case of the elliptic plane there is a finite number of essentially
different kinds of fundamental regions, and a finite number of congruent
regions suffices for a complete covering of the whole plane; the group
consists indeed of a finite number of motions only. In the case of the
hyperbolic plane there is an infinite number of essentially different
kinds of fundamental regions, namely, the well-known Poincaré polygons.
For the complete covering of the plane an infinite number of congruent
regions is necessary. The case of Euclid's plane stands between these;
for in this case there is only a finite number of essentially different
kinds of groups of motions with fundamental regions, but for a complete
covering of the whole plane an infinite number of congruent regions is
necessary. Exactly
the corresponding facts are found in space of three dimensions. The fact
of the finiteness of the groups of motions in elliptic space is an
immediate consequence of a fundamental theorem of C. Jordan,40 whereby the number of essentially different kinds of finite groups of
linear substitutions in n variables does not surpass a certain
finite limit dependent upon n. The groups of motions with
fundamental regions in hyperbolic space have been investigated by Fricke
and Klein in the lectures on the theory of automorphic functions,41
and finally Fedorov,42
Schoenflies43
and lately Rohn44
have given the proof that there are, in euclidean space, only a finite
number of essentially different kinds of groups of motions with a
fundamental region. Now, while the results and methods of proof
applicable to elliptic and hyperbolic space hold directly for n-dimensional
space also, the generalization of the theorem for euclidean space seems
to offer decided difficulties. The investigation of the following
question is therefore desirable: Is there in n-dimensional euclidean
space also only a finite number of essentially different kinds of groups
of motions with a fundamental region? A
fundamental region of each group of motions, together with the congruent
regions arising from the group, evidently fills up space completely. The
question arises: whether polyhedra also exist which do not appear as
fundamental regions of groups of motions, by means of which nevertheless
by a suitable juxtaposition of congruent copies a complete filling up of
all space is possible. I point out the following question, related to
the preceding one, and important to number theory and perhaps sometimes
useful to physics and chemistry: How can one arrange most densely in
space an infinite number of equal solids of given form, e. g.,
spheres with given radii or regular tetrahedra with given edges (or in
prescribed position), that is, how can one so fit them together that the
ratio of the filled to the unfilled space may be as great as possible? If
we look over the development of the theory of functions in the last
century, we notice above all the fundamental importance of that class of
functions which we now designate as analytic functions--a class of
functions which will probably stand permanently in the center of
mathematical interest. There
are many different standpoints from which we might choose, out of the
totality of all conceivable functions, extensive classes worthy of a
particularly thorough investigation. Consider, for example, the class
of functions characterized by ordinary or partial algebraic differential
equations. It should be observed that this class does not contain the
functions that arise in number theory and whose investigation is of the
greatest importance. For example, the before-mentioned function
will have to be
used. If,
on the other hand, we are lead by arithmetical or geometrical reasons to
consider the class of all those functions which are continuous and
indefinitely differentiable, we should be obliged in its investigation to
dispense with that pliant instrument, the power series, and with the
circumstance that the function is fully determined by the assignment of
values in any region, however small. While, therefore, the former
limitation of the field of functions was too narrow, the latter seems to
me too wide. The idea of the analytic function on the other hand includes
the whole wealth of functions most important to science whether they have
their origin in number theory, in the theory of differential equations or
of algebraic functional equations,whether they arise in geometry or in
mathematical physics; and, therefore, in the entire realm of functions,
the analytic function justly holds undisputed supremacy. |
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