Slides of presentation (PDF, 341 Kb)
Joint work with Daniel Lichtblau.
We study a question with connections to
linear algebra, real algebraic geometry, combinatorics, and complex analysis.
Let $p(x,y)$ be a polynomial of degree $d$ with $N$ positive coefficients and
no negative coefficients, such that $p=1$ when $x+y=1$. It is known that the
sharp estimate $d \leq 2N-3$ holds. We study the $p$ that minimize $N$ and
we give complete classification of these polynomials up to $d=17$ by
computational methods. We use separately a linear algebra approach and a
mixed-integer programming approach. The question is motivated by a problem
in CR geometry. In particular, a complete classification of polynomials
minimizing $N$ is an important first step in the complete classification of
CR maps of spheres in different dimensions. |