# Newton-Okounkov Bodies and Khovanskii Bases

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Kiumars Kaveh University of Pittsburgh | Kiumars Kaveh University of Pittsburgh | ||

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+ | Chris Manon George Mason University | ||

Frank Sottile Texas A&M University | Frank Sottile Texas A&M University |

## Revision as of 03:36, 22 December 2016

## **Overview**

The theory of Newton-Okounkov bodies generalizes that of Newton Polytopes, giving a conceptual framework for root counts to systems of functions in terms of volumes and mixed volumes of convex Newton-Okounkov bodies. While the asymptotic definition of these bodies is not particularly constructive, when they admit a Khovanskii basis, they are polyhedral. Having a Khovanskii basis enables other methods based on polyhedra to be used.

While Newton-Okounkov bodies arose to solve questions from pure mathematics they have significant potential in applications. These include providing a foundation for root counts for polynomial systems from applications, the use of these root counts and Khovanskii bases for solving, and a host of algorithmic questions involving computing/determining Newton-Okounkov bodies and Khovanskii bases. The purpose of this minisymposium is to explore some of these opportunities and to advertise this to the wider community of applied algebraic geometry.

## **Organizer**

Frank Sottile

## **Confirmed Speakers**

Lara Bossinger Universitaet Koeln

Kiumars Kaveh University of Pittsburgh

Chris Manon George Mason University

Frank Sottile Texas A&M University