by Gerald Schmieder

The increasing specialisation in all scientific fields means that the questions and results of current research are difficult to communicate to laypeople and even non-experts in their own field. This is particularly true for mathematics, because its research subjects are hardly accessible to the senses. However, this is not generally the case. In addition to general comments on mathematical research in the past and today, the "Sendov conjecture" will be presented as an example of current research, the understanding of which does not require any previous mathematical training beyond school knowledge.

Is there still something to research in the old science of mathematics? If at all, so the preconception goes, such activity ekes out an anaemic shadowy existence far removed from any relevance. The presentation of mathematical methods and facts at school and university gives the impression of a self-contained view of the world. The methods available seem to be completely sufficient to solve all mathematical problems. What's more, today's computers do away with tedious calculations.

What is the reality of mathematical research? It is not difficult to ask mathematical questions that can be formulated using elementary school knowledge, but which cannot yet be answered using all the methods available.

Only recently, the famous conjecture of Pierre de Fermat (1601 - 1665), a problem more than 300 years old, was proved by the English mathematician Andrew Wiles, a mathematical research result that was also reported in the world press. What do we gain from it? For everyday life, it is of course irrelevant to know that the equation xn + yn = zn for n = 3, 4, 5, ... has no non-zero integer solutions x, y, z, as Fermat claimed. However, in dealing with this problem, essential methods of modern algebra have been developed, which have already been put into practice in cryptography.

The demand Application Now! for immediate application is therefore no more applicable to mathematical research than it is to the planting of oak trees. Unsolved problems, unless they are generally unsolvable, are a sign of a lack of adequate methods. Finding them is almost always more important than answering the question that led to their search.

For example, differential calculus, which is indispensable today, owes its origin to the problem of strictly defining the concept of a tangent. Before Isaac Newton (1642 - 1727) and Gottfried W. Leibniz (1646 - 1716), others had already worked on this problem, including René Descartes (1596 - 1650), who called it the "most general and most useful problem" that he knew. Today, everyday life cannot do without differential calculus, which not only allows various little men in our households to function.

There are often limits to the explicit solution of a mathematically formulable problem in a specific case. In the following example, it is the large number of parameters to be taken into account. In such cases, not even a computer can help.

Problem 1: Consider four squares of edge length 1, which are to be distributed in the plane in such a way that they touch each other but do not overlap. We define the diameter of each square configuration as the greatest distance between two points of the four squares. In which configuration is the diameter absolutely minimal?

The assumption is, of course, that the diameter is smallest when the four squares are arranged in such a way that they join together to form a square with an edge length of 2. However, no one has yet found a proof. The conjecture certainly has a high level of evidence, but this is no substitute for a real proof (as the next problem clearly shows). However, it is unlikely that a rigorous solution to this problem will ever be found. I dare to predict that a revolution in mathematical methods would have to accompany it, similar to the discovery of infinitesimal calculus by Leibniz and Newton.

Aren't mathematically exact solutions superfluous educational ballast, from which practitioners today can fortunately free themselves by applying "empirical" methods and thus gain time for more important things? The next example clearly shows what we should think of such simplifications.

Problem 2: Let n different points be chosen on the edge of a circular disc, all of which are connected to each other in a straight line. How many parts will the circular disc be cut into?

If we use Gn to denote the number of sections for n given edge points, we can determine G1 = 1, G2 = 2, G3 = 4, and <g4></g4>by trial and error. Thus, Gn = 2n-1 seems to be certain knowledge.

But this is not true. Although G5 = 16 strengthens the hypothesis again, the disillusionment comes with six points: G6 = 31 (if three connecting lines meet at a common point: G6 = 30), but not 32!

In reality, the equation for Gn is not so obvious. Finding it has the status of a difficult maths exercise.

Questions of this type are generally known: Given the beginning of a sequence: a1, a2, a3, a4, a5. How big is a6? Such questions appear in intelligence tests, for example. What do we make of this? If, for example, 1, 2, 4, 8, 16 are presented, it is simply wrong to accept 32 as the "only intelligent" continuation.

We will now present a problem that was published 35 years ago but is still unresolved despite many efforts to prove and disprove it: Sendov's conjecture. The Bulgarian mathematician Blagovest Sendov was introduced to questions about the distribution of zeros by his teacher Nikola Obreschkoff. He has been politically active in high positions for several years, e.g. he was President of the Bulgarian Parliament. However, he has always maintained contact with the mathematical world (a meeting with Boris Yeltsin on the fringes of a mathematics congress in Moscow in 1996 led to irritation in political circles in Sofia, see www.b-info.com/places/Bulgaria/news/96-02/feb01.bta). In addition to the purely mathematical formulation, Sendov's conjecture also allows a physical description, which we will begin with here.

Problem 3a: In a plane, let there be a finite number of electrons, but at least two, fixed at different locations within a circular disc of radius 1. In the generated force field there are points of rest, i.e. places where no force is exerted on a test charge. The assumption is now: The nearest point of rest to each of the electrons involved has a distance of at most 1.

The mathematical formulation of the problem requires knowledge of complex numbers, which will be presented here briefly and in a generally understandable way. The term "complex number" was introduced by Carl Friedrich Gauss (1777 - 1855) and first appears in his treatise Theoria residorum biquadraticorum from 1831. A plane is considered in which a point is defined as the zero point and a half-straight line ("positive x-axis") starting at the zero point. As is well known, the points of such a plane can be described by specifying two real numbers, the so-called coordinates (remember: the real numbers are the positive or negative numbers with any number of decimal places). This is usually done in the form of Cartesian coordinates named after Descartes, but sometimes the so-called polar coordinates are more favourable. A point P in this plane is described by specifying its distance rP to the zero point and the angle aP that the connecting line zero point - point encloses with the positive x-axis.

Addition and multiplication are now to be introduced for the points in the plane. The addition is explained as the usual vector addition. The product P×Q of the points P and Q with the polar coordinates (rP, aP) and (rQ, aQ) is the point with the polar coordinates (rP.rQ,aP+aQ). The arithmetic operations defined in this way fulfil the usual arithmetic rules.

The plane with these arithmetic operations forms the complex numbers. Those points of the plane that belong to the angle 0° or 180° have a special position. These points remain among themselves with regard to both addition and multiplication, which results from the definition. If we identify the points(r, 180°) and(r, 0°) with the real numbers -r and r respectively, we can see that the addition and multiplication just explained results in the same as the corresponding calculation with the assigned real numbers.

So we find the real numbers "disguised" as part of the complex numbers. Once we have recognised this, we consider the real numbers as part of the complex number plane. Two and a half centuries passed before the completely concrete and vivid interpretation described here, which was first given in 1799 by Kasper Wessel (1745 - 1818). Since its first appearance in 1545 (Geronimo Cardano (1501 - 1576): De Regula falsum ponendi), the complex numbers had been recognised as useful, but because of

i2 = (1, 90°).(1,90°) = (1,180°) = -1

"non-existent entities in reality" (i, the usual designation of (1, 90°), stands for "imaginary"), an argumentative quagmire.

We now define a (normalised) polynomial as a function described by an equation of the form where a0, a1,..., an-1 are fixed (real or complex) numbers and z stands for the variable. The highest power n occurring in a polynomial is called the degree of the polynomial. For such a polynomial, the complex numbers form the natural range of the variable z, not the real numbers. According to the so-called "fundamental theorem of algebra", there are complex numbers z1, ..., zn , so that p can also be written as a product for all complex numbers z. Since the principle "a product is exactly 0 if at least one factor is 0" also applies to complex numbers, this is called: A polynomial of the above form has exactly n zeros in the complex numbers.

The derivative p' of the complex polynomial p can be defined in the same way as for real functions. The same derivation rule applies to polynomials, so to understand the following text it is sufficient to take this equation as the definition of the derivative p' of the polynomial p without any further meaning.

Everything is now ready for the mathematical formulation of Sendov's conjecture.

Problem 3b: If p is any complex polynomial (of degree at least 2) whose zeros are all at most 1 away from the origin, then Sendov's conjecture states that the distance from each zero of p to the nearest zero of p'> at most 1.

The zeros of the polynomial in the mathematical formulation correspond to the electrons in the physical formulation and the derivative zeros correspond to the rest points. A number of partial results have been obtained for Sendov's conjecture. It is known, for example, that it is correct for polynomials with degrees up to and including 7. However, the methods of proof are hardly extendable.

In the following, we will primarily examine the question of why a proof of this conjecture is so difficult to find. By considering "movements" of the zeros and the derivative zeros, two principles can be made heuristically plausible, which are also easy to understand in the physical model of electrons and rest points:

(I) If some zeros of p are sufficiently close to each other, there will be (at least) one zero of p' in the collection.

(II) In the vicinity of a solitary zero of p there is a zero of p'.

However, it is very difficult to summarise these principles qualitatively and to specify "near" and "far". Corresponding details from the author's work cannot be discussed here.

It seems to be the case that Sendov's assumption cannot be justified by one of the two principles alone, but is based on an interplay between the two: If there are others in sufficient proximity to the zero zj of p just considered, then (I) provides what is desired; if this is not the case, then perhaps zj is isolated enough to allow (II) to take hold, thus guaranteeing the sought-after derivative zero. Qualitative specifications are not yet sufficient to complement Sendov's conjecture in the sense described above.

Perhaps the assessment that Sendov's conjecture can be proved from the two principles is too optimistic. In fact, it is not beyond imagination that it is wrong. However, it cannot be very wrong. To understand this, let us ask the question in a more general form:

Problem 3c: If p is an arbitrary complex polynomial (of degree at least 2) whose zeros are all at most 1 away from the origin, how large can the distance r of each individual zero to the next derivative zero be at most?

Sendov's conjecture is then reduced to the assertion"r = 1". It seems clear that the statement is correct for r = 2, as there can be no point of rest of the field outside the circular disc K, as the acting forces cannot cancel each other out here. However, as plausible as this may seem, the proof for the mathematical statement " p' has all zeros in the circular disc" is by no means trivial; it was given by F. Lucas in 1874. A few decades earlier, Gauss had already had the idea for this theorem and noted it in his notebook, albeit without a proof. However, the current state of knowledge is far more advanced than r = 2. It is now known that the statement is already true for r = 1.084. The correctness of Sendov's conjecture is therefore "almost" proven.

We can conclude that the existing methods are not bad, but they are not yet sufficient to solve the problem at hand. So there is still some work to be done. A future complete proof of Sendov's conjecture would certainly be a small step, if only the numerical progress achieved from r = 1.084 to r = 1 is considered. But it could certainly be "a big step for mankind" if the study of this question were to produce new, effective methods, which would then bear fruit elsewhere and probably only much later.

Prof Dr Gerald Schmieder (50) has been teaching and researching at the University of Oldenburg since 1990. He studied mathematics and physics at the University of Hanover. After his habilitation, he spent time at the Université de Montréal (Canada). Before he was appointed to Oldenburg, he taught in Hanover and Würzburg. His main field of work is function theory, to which Sendov's conjecture also belongs. He has been Dean of the Department of Mathematics since April 1997. In addition to his university teaching activities, he is a passionate violinist. He was concertmaster of the university orchestra for several years.