Keynote speaker Bhargava emphasizes polynomials

Last Saturday, the College hosted the 20th annual Hudson River Undergraduate Mathematics Conference (HRUMC).  HRUMC is a one-day conference that attracts students and faculty from many institutes of higher education throughout New England.

At the conference, students and faculty gave short talks ranging from original research to expository talks. The conference was designed to give students experience in a professional mathematics setting.

The conference featured over 150 talks scheduled throughout the day.  There was a morning session  following registration, after which the keynote speaker, Dr. Manjul Bhargava, gave an address.

After lunch, participants attended an afternoon session.  Around 500 student and faculty members in total attended the conference, nearly filling the MainStage at the ’62 Center for Theatre and Dance, for Bhargava’s talk.

Bhargava, a highly acclaimed number theorist from Princeton, gave the keynote address titled “How likely is it for a polynomial to take on a square value?” He pointed out the importance of answering this query, saying: “This is a very classical question in mathematics. It’s very important to understand polynomials.”

Bhargava began his talk by examining ancient questions concerning when polynomials take on square values.

Around 2500 B.C., the Egyptians built megalithic monuments that contained right triangles with integer lengths.  They wanted to solve the equation a2+b2=c2 so that a, b, and c all took on integer values. Although they did find solutions, they didn’t write them down but simply carved them in stone.

In 800 B.C., there was a Mesopotamian tablet made called the Plimpton 322. The tablet listed a, b and c values which were integers that worked in the equation a2+b2=c2. “They seemed to have a method,” Bhargava said.  Unfortunately, Plimpton 322 only contained a list of values, not the method used to find them.

Bhargava then moved onto a problem posed by Srinivasa Ramanujan in 1913.  He claimed that n=3, 4, 5, 7, 15 are the only solutions that cause 2n-7 to take on a square value.

“He found these solutions and asked people to prove that,” Bhargava said, eliciting laughs from the audience.  It was eventually solved by Trygve Nagell in 1948, but “the problem continues to generate research,” according to Bhargava.

After discussing Ramanujan’s problem further, Bhargava then moved into the crux of his talk.

“Suppose we just take a polynomial in one variable. When does it take a square value?” Bhargava asked.  This involves looking for points on the curve y2=f(x), where f(x) is  a polynomial whose coordinates are integers.  Number theorists often begin solving this problem by looking for rational solutions first.

Generally, polynomials are graphed in two dimensions, over real numbers.  However, it makes more sense to graph polynomials in four dimensions, over the complex numbers.

“We don’t have 4-D paper, though,” Bhargava said.  Fortunately, when graphed in four dimensions, the result is a surface. “It’s not too hard to visualize because in the end it is a surface, which is two dimensional,” Bhargava said.

Bhargava then moved into the more technical part of his speech, talking about the genus of a surface, which is the number of holes occurring in a topological surface, and how the genus reveals information about whether the polynomial has zero rational solutions, a finite number, or infinitely many.  Because the genus is related to the degree of a polynomial, it is useful to examine the case of elliptic curves, graphs of polynomials of degree three, and when they take on square values.

The genus of these curves is one, which means that they either have finitely many rational points or infinitely many.

Through Bhargava’s work with Arul Shankar and Chris Skinner, he has proven that a positive proportion of elliptic curves have infinitely many rational points and a positive proportion have no rational points.  He expects that 50 percent will take on square values and 50 percent will never take on square values.  This result holds for degree four polynomials.  He also expects that when the degree is more than five, polynomials have no rational points that take on square values.

Bhargava’s talk concluded with a short Q-and-A session.