Lorentzian polynomials from polytope projections

“ Polytope Projections ” ( Discrete Geometry

Barn Raisings of Four-Dimensional Polytope Projections

By constructing large, attractive polytope models, students or a general audience are connected with the beauty of mathematics in a hands-on manner. Four-dimensional geometry provides engaging challenges to think about while working in a team to build a physically impressive, sculptural result. I have led many workshops of this sort with Zometool materials and report on one event in detail. A f...

Affine projections of polynomials

An m-variate polynomial f is said to be an affine projection of some n-variate polynomial g if there exists an n×m matrix A and an n-dimensional vector b such that f(x) = g(Ax + b). In other words, if f can be obtained by replacing each variable of g by an affine combination of the variables occurring in f , then it is said to be an affine projection of g. Given f and g can we determine whether...

Random Polynomials with Prescribed Newton Polytope

The Newton polytope Pf of a polynomial f is well known to have a strong impact on its behavior. The Kouchnirenko-Bernstein theorem asserts that even the number of simultaneous zeros in (C∗)m of a system of m polynomials depends on their Newton polytopes. In this article, we show that Newton polytopes further have a strong impact on the distribution of mass and zeros of polynomials, the basic th...

On polynomials in two projections

Conditions are established on a polynomial f in two variables under which the equality f(P1, P2) = 0 for two orthogonal projections P1, P2 is possible only if P1 and P2 commute.