## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

Wednesday, July 3, 2019 — 10:00 AM EDT

**Anton Mosunov, Department of Pure Mathematics, University of Waterloo**

"Generalizations of the Gap Principle and the Thue-Siegel Principle, With Applications to Diophantine Equations"

We develop generalizations of two well-known principles from the theory of Diophantine approximation, namely the gap principle and the Thue-Siegel principle. Our results find their applications in the theory Diophantine equations. Let $\alpha$ be an algebraic number over $\mathbb Q$ and let $F(X, Y)$ be the homogenization of the minimal polynomial of $\alpha$. In the special case when $\mathbb Q(\alpha)/\mathbb Q$ is a Galois extension of degree at least seven, we establish absolute bounds on the number of solutions of certain equations of Thue and Thue-Mahler type, which involve $F(X, Y)$. Consequently, we give theoretical evidence in support of Stewart's conjecture (1991). More generally, if every conjugate $\beta$ of $\alpha$ is such that the degree of $\beta$ over $\mathbb Q(\alpha)$ is small relative to the degree of $\alpha$ over $\mathbb Q$, we establish bounds of the form $C\gamma$, where $C$ is an absolute constant and $\gamma$ is a natural parameter associated to $\alpha$ that does not exceed the degree of $\alpha$ over $\mathbb Q$. We expect this parameter to be small, perhaps even bounded by an absolute constant.

MC 2009

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Office of Indigenous Relations.