40 Normat 51:1, 40 (2003)

Summary in English

Khadija Laghrida Christensen and

Ole Christensen, Linear independence

in function spaces. (Danish.) This ar-

ticle considers vector spaces generated

by some special functions, and exam-

ines whether their linear combinations

have unique representations. The au-

thors start with the most elementary

cases, namely, polynomials and trigono-

metric functions. Next they consider

complex exponential functions and some

more complicated systems of functions

which have recently attracted much at-

tention in pure mathematics as well as

in applied science. Finally, some open

problems related to those systems are

presented. These are easy to formulate,

but apparently very diﬃcult to solve.

Kent Holing, When does the quartic

equation have constructible roots? (Nor-

wegian.) This article provides a simple

criterion for counting the number of con-

structible roots of a quartic equation

with rational coeﬃcients. The answer is

given in terms of the number of ratio-

nal roots of the resolvent equation. After

some examples and exercises, the arti-

cle ends by connecting this result to the

computation of Galois groups of quartic

polynomials.

Gert Almkvist, Strings in moonshine

I. (Swedish.) The author considers vari-

ous special power series and Lambert se-

ries with integer coeﬃcients, that arise

from linear diﬀerential equations. Some

of these curious series also turn up in

algebraic geometry and representation

theory, where their coeﬃcients count the

numb e r of lines on particular algebraic

varieties or the dimensions of the rep-

resentations of particular groups. Here

the topic is approached from the point of

view of diﬀerential equations. The con-

straint of integer coeﬃcients turns out to

be very tight. For one important class of

equations of the fourth order, only four-

teen such cases are known. The author

presents one of these cases, wh ich pos-

sesses an unexpected link to algebraic

geometry.

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