Mathematicians Resurrect Hilbert’s 13th Problem
Success is rare in math. Just ask Benson Farb.
“The hard part about math is that you’re failing 90% of the time, and you have to be the kind of person who can fail 90% of the time,” Farb once said at a dinner party. When another guest, also a mathematician, expressed amazement that he succeeded 10% of the time, he quickly admitted, “No, no, no, I was exaggerating my success rate. Greatly.”
Farb, a topologist at the University of Chicago, couldn’t be happier about his latest failure — though, to be fair, it isn’t his alone. It revolves around a problem that, curiously, is both solved and unsolved, closed and open.
The problem was the 13th of 23 then-unsolved math problems that the German mathematician David Hilbert, at the turn of the 20th century, predicted would shape the future of the field. The problem asks a question about solving seventh-degree polynomial equations. The term “polynomial” means a string of mathematical terms — each composed of numerical coefficients and variables raised to powers — connected by means of addition and subtraction. “Seventh-degree” means that the largest exponent in the string is 7.
Mathematicians already have slick and efficient recipes for solving equations of second, third, and to an extent fourth degree. These formulas — like the familiar quadratic formula for degree 2 — involve algebraic operations, meaning only arithmetic and radicals (square roots, for example). But the higher the exponent, the thornier the equation becomes, and solving it approaches impossibility. Hilbert’s 13th problem asks whether seventh-degree equations can be solved using a composition of addition, subtraction, multiplication and division plus algebraic functions of two variables, tops.
The answer is probably no. But to Farb, the question is not just about solving a complicated type of algebraic equation. Hilbert’s 13th is one of the most fundamental open problems in math, he said, because it provokes deep questions: How complicated are polynomials, and how do we measure that? “A huge swath of modern mathematics was invented in order to understand the roots of polynomials,” Farb said.