In a recent paper, Manjul Bhargava of Princeton University has solved an 85-year-old question about the solutions to polynomial equations such as x 2 and 3x.
The values of x that make the polynomial equal zero are studied by mathematicians. You will get zero if you plug the number 1 or 2 into x 2.
The equation x 2 is difficult to understand. A rational number can be used to solve the polynomial. A new number is defined, and mathematicians call it $latexsqrt5$. We don't know anything about the square of $latexsqrt5$. You can easily get a second root with the multiplication of $latexsqrt5$.
The two equations are different in a critical way. The roots of x 2 – 5 are important in helping solve many equations in our mathematical system. Our mathematical system is limited to rational numbers. If we start using them this way, we will find that they are completely interchangeable. Both $latex 2sqrt5$ and x 2sqrt5$ work equally well. It is helpful to anywhere $latex sqrt5$.
The most common scenario is this one. If you start using 2 in place of 1 in your mathematical system, it will be nonsense.
The Dutch mathematician Leendert van der Waerden formulated the van der Waerden conjecture in 1936 to try to quantify how many polynomials have non interchanging roots. For decades, progress was slow. More concrete, classical questions in number theory made a comeback in the past 20 years.
Bhargava, who received the Fields Medal, invited people to go exploring.
There was a lot of new work on the van der Waerden conjecture in the summer of 2021. A new paper made significant headway on June 28 after Dietmann and his partner Sam Chow solved a few key cases. A team of six shared their own preprint.
Bhargava gave an online talk on July 1. Bhargava presented a proof of a slightly modified van der Waerden conjecture.
Bhargava shared new work proving van der Waerden's conjecture in its entirety in an online presentation during the Mathematical Congress of the Americas. He posted his paper online.
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