Two years ago, Boris Alexeev emailed me a problem:

Let . Suppose you have distinct numbers in some field. Is it necessarily possible to arrange the numbers into an matrix of full rank?

Boris’s problem was originally inspired by a linear algebra exam problem at Princeton: Is it possible arrange four distinct prime numbers in a rank-deficient matrix? (The answer depends on whether you consider to be prime.) Recently, Boris reminded me of his email, and I finally bothered to solve it. His hint: Apply the combinatorial nullstellensatz. The solve was rather satisfying, and if you’re reading this, I highly recommend that you stop reading here and enjoy the solve yourself.

Continue reading A neat application of the polynomial method