This is an original gift, to say the least, offered by the mathematician Manjul Bhargava to his colleague Don Zagier for his birthday: a demonstration of van der Waerden’s conjecture. “In the mathematical community, we like to celebrate birthdays, says, with a smile, Manjul Bhargava, professor at Princeton University and winner of the Fields Medal in 2014. I know that Don Zagier likes this kind of very concrete fundamental questions, so I was particularly happy to be able to present this proof at the occasion of [la conférence en l’honneur de] son 70e anniversary ! » This atypical gift consists of a few dozen pages that offer a clever solution to a mathematical problem that is more than 80 years old.
This problem is part of the continuity of a very long mathematical quest: the “solvability by radicals of polynomials”. Polynomials are those famous functions that can be written, for example, f (x) = ax2 + bx + c. All high school students in France learn in mathematics lessons to calculate the “roots” of polynomials like f — which are said to be “second degree” — i.e. the values of x for which f (x) is 0. But if we consider more complicated polynomials, studying their roots becomes very difficult!
A result demonstrated in the 19the century by the Norwegian mathematician Niels Henrik Abel (1802-1829) ensures that “Polynomials of degree greater than or equal to five are generally not solvable by radicals”. Translation: unlike the case of second-degree polynomials studied in high school, there is no “simple” formula for finding the roots of a polynomial of degree five or more.
A few years after Abel, the French mathematician Evariste Galois (1811-1832) made the connection between the solvability of polynomials by radicals and a particular mathematical object – which would later be called the “Galois group” in his honor. He understands that the more the Galois group associated with a given polynomial is « gros »the more difficult it is to determine the roots of the latter.
Study symmetries
It is here that the famous conjecture comes into play, of which Manjul Bhargava has just proposed a demonstration. In 1936, the Dutch mathematician Bartel Leendert van der Waerden (1903-1996) put forward a hypothesis on the proportion of polynomials whose Galois group is “as big as possible”. In essence, he conjectures that there are many. Since then, no one had come up with a complete demonstration.
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