## Departmental PhD Thesis Exam – Leonid Monin

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Friday, April 26, 2019
11:10 a.m.
BA6183

PhD Candidate:  Leonid Monin
Thesis title: Newton Polyhedra, Overdetermined system of equations, and Resultants

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In the first part of this thesis we develop Newton polyhedra theory for overdetermined systems of equations. Let A_1…A_k be finite sets in Z^n  and let be an algebraic subvariety of (C^*)^n  defined by a system of  Laurent polynomials with supports in A_1… A_k.  Assuming that  Laurent polynomials are sufficiently generic, the Newton polyhedron theory computes discrete invariants of Y in terms of their Newton polyhedra. It may appear that the generic system with fixed supports  is inconsistent. In this case one is interested in the generic consistent system. We extend Newton polyhedra theory to this case and compute discrete invariants generic non-empty zero sets. Unlike the classical situation, not only the Newton polyhedra of Laurent polynomials, but also their supports themselves appear in the answers.

We proceed then to the study of overdetermined collections of linear series on algebraic varieties other than algebraic torus. That is let E_1…E_k be a finite dimensional subspace of the space of  regular sections of line bundles on an irreducible algebraic variety X, so that the system

s_1 = … = s_k = 0,

where s_i is a generic  element of E_i does not have any roots on X. In this case we investigate the consistency variety  (the closure of the set of all systems which have at least one common root) and study general properties of zero sets Z of a generic consistent system. Then, in the case of equivariant linear series on spherical homogeneous spaces we provide a strategy for computing discrete invariants of such generic non-empty set Z.

The second part of this thesis is devoted to the study of Delta-resultants of (n+1)-tuple of Laurent polynomials with generic enough Newton polyhedra.  We provide an algorithm for computing Delta-resultant assuming that an n-tuple f_2, …, f_{n+1} is developed. We provide a relation between the product of f_1 over roots of  f_2 = … = f_{n+1} = 0 in (C^*)^n and the product of f_2 over roots of f_1=f_3 = … = f_{n+1} = 0 in (C^*)^n assuming that the n-tuple (f_1f_2, f_3…f_{n+1} is developed. If all n-tuples contained  in (f_1…f_{n+1}) are developed we provide a signed version of Poisson formula for Delta-resultant. Interestingly, the sign of the sparse resultant is nontrivial and is defined through Parshin symbols. Our proofs are based on a topological version of the Parshin reciprocity laws.
A copy of the thesis can be found here:  ut-thesis monin