Certifying that a given polynomial function cannot take negative values is useful in the context of many mathematical and scientific applications.
One way to address this certification problem is to understand the combinatorial structure underlying the support of the polynomial that we are interested in.
Given a polynomial $f$, the support of $f$ consist of all of its exponent vectors, and the Newton polytope associated to $f$ is the convex hull of its support.
The support of the polynomial $f$ yields a signed and weighted point configuration whose points are in the Newton polytope of $f$.
The main objective of this talk is to present some known and novel techniques that exploit this combinatorial information about the polynomial $f$, in order to draw conclusions about the nonnegativity of $f$.
The talk is structured in three main sections with increasing difficulity. The first section is dedicated to a brief revision of some definitions and facts about polynomials and polyhedral geometry.
In the second part of the talk, we present two techniques to decide the nonnegativity using the information encoded in the aforementioned point configuration. In the last section, we show some explicit applications of these techniques to detect the existence of multiple solutions in certain ODE systems arising from the chemical reaction networks theory.

Link to the seminar is:

https://teams.microsoft.com/l/meetup-join/19%3a0ae91f7b86a24e8fa1d818a6f74b22a1%40thread.tacv2/1649141048717?context=%7b%22Tid%22%3a%22066690f2-a8a6-4889-852e-124371dcbd6f%22%2c%22Oid%22%3a%22a343f6fd-86f8-4abe-95cf-3c7f4ad5f0ca%22%7d