Mathematicians resurrect Hilbert's 13th problem

David Hilbert's question about polynomials of the seventh degree, which was considered solved for a long time, opened a new network of mathematical connections for researchers

Success in mathematics is rare. Just ask Benson Farb.

“The problem with mathematics is that 90% of the time you fail, and you have to be the person to accept it,” Farb once said over dinner with friends. When one of the guests, also a mathematician, was surprised that Farb was successful as much as 10% of the time, Farb admitted: “No, no, I greatly exaggerated my success rate.”

Farb, a topologist at the University of Chicago, is happy to meet his latest setback - though, in all fairness, it's not entirely his credit. The question is connected with a problem, paradoxically solved and unresolved at the same time, open and closed. The problem is the 13th of 23 mathematical problems that were not solved at the beginning of the 20th century. Then the German mathematician David Hilbert compiled this list, which, in his opinion, determined the future of mathematics. The problem is related to solving polynomial equations of the seventh degree. A polynomial is a sequence of terms of an equation, each of which consists of a numerical coefficient and variables raised to a power; terms are connected with each other by addition and subtraction. The seventh degree means the largest exponent of all variables.

Mathematicians have already learned to skillfully and quickly solve equations of the second, third and, in some cases, fourth order. These formulas - including the familiar quadratic formula for the second degree - include algebraic operations, that is, arithmetic operations and extraction of roots. But the larger the exponent, the more confusing the equation, and it becomes more and more difficult to solve. Hilbert's 13th problem is the question of whether it is possible to express the solution of a seventh-order equation in terms of a set of additions, subtractions, multiplications, divisions and algebraic functions from at most two variables.

Answer: probably not. For Farb, however, this is not just a matter of solving a complex algebraic equation. He said that Problem 13 is one of the most fundamental problems in mathematics, as it raises deep questions: How complex are polynomials, and how can they be measured? “A whole layer of modern mathematics has been invented to better understand the roots of polynomials,” Farb said.

This problem pulled him and the mathematician Jesse Wolfson of the University of California at Irvine down the mathematical rabbit hole, the moves of which they still study. She also brought in Mark Kissin, a Harvard number theorist and an old friend of Farb's, to their excavation.

Farb admitted that they have not yet solved Hilbert's 13th problem, or even come close to solving it. However, they unearthed nearly extinct mathematical strategies, and explored the problem's links to various fields of knowledge, including complex analysis, topology, number theory, representation theory, and algebraic geometry. They applied their own approaches, in particular, combining polynomials with geometry and narrowing the range of possible answers to Hilbert's question. Their work also proposes a method for classifying polynomials by complexity metrics - an analogue of complexity classes related to the unsolved problem of equality of the classes P and NP.

“They were actually able to extract a more interesting version of interest from interest” compared to those studied previously, said Daniel Litt, a mathematician at the University of Georgia. “They show the mathematical community a lot of natural and interesting questions.”

Opened, closed and reopened

Many mathematicians already thought the problem was solved. In the late 1950s, the brilliant Soviet scientist Vladimir Igorevich Arnold and his mentor Andrei Nikolaevich Kolmogorov published their proofs. For most mathematicians, the work of Arnold-Kolmogorov closed this question. Even on Wikipedia - not the ultimate truth, but a rather intelligent broker in the search for knowledge - until recently the problem was marked as solved.

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