Who invented Hilbert's basic theorem

Polynomials: Hilbert's 13th problem

You seldom achieve success in math. "The hard thing about the subject is that 90 percent of the time you fail, and you have to be the kind of person who can handle it," Benson Farb once said at a dinner party. When another guest, also a mathematician, asked in surprise whether he was really successful in one out of ten cases, he quickly admitted: “No, no. I exaggerated my quota - very much. "

Farb, a topologist at the University of Chicago, couldn't be happier with his recent failure. It concerns a centuries-old problem that, curiously, is both solved and unsolved. In his famous 1900 speech at the International Congress of Mathematicians, David Hilbert placed the problem 13th of the 23 major problems that he predicted would shape the century to come. It has to do with solving polynomial equations of the seventh degree.

This article is included in Spectrum - The Week, 07/2021

Polynomials are made up of coefficients and exponentiated variables that are connected by addition and subtraction. The degree corresponds to the highest exponent that appears in it. Mathematicians have already developed sophisticated methods to solve the zeros of polynomials of the second, third and sometimes fourth degree. These formulas (such as the p-q formula known from school or the midnight formula) consist of algebraic operations, i.e. addition, multiplication, subtraction, division and the extraction of the square root. The higher the exponent, the more bulky a solution formula - if it exists at all.

Up to the fourth grade

Because such a general formula can only be given for polynomials up to the fourth degree. In order to calculate the zeros of such equations with higher exponents, one has to resort to other methods. Nevertheless, the solutions can be determined by algebraic operations paired with functions that are also algebraic and depend on several parameters. Hilbert's 13th problem revolves around the question of whether one can solve polynomials of the seventh degree using algebraic functions that are determined by only two variables.

Probably the answer to that is no. For Farb, however, it's not just about solving an equation. According to him, it is one of the most fundamental open problems in the field because it leads to further profound questions: How complicated are polynomials and how can they be determined? “A large part of modern mathematics was invented to understand the zeros of such equations,” explains Farb.

The problem never left him and his colleague Jesse Wolfson of the University of California, Irvine, and they are still researching it today. They also brought in Mark Kisin, a good friend of Farb who is a number theorist at Harvard University. However, they are far from a solution, admits Farb.