## 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

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Thursday, October 23, 2014 — 1:30 PM EDT

In this talk, we will present the results from the recent arXiv paper by A. Dubickas, K. G. Hare and J. Jankauskas on the solution of three and four term linear equations in the conjugates of a Pisot number.

More precisely, we show that the number a=(1+\sqrt{3+2\sqrt{5}})/2 with minimal polynomial x^4-2x^3+x-1 is the only Pisot number whose four distinct conjugates a_1, a_2, a_3, a_4 satisfy the additive relation a_1+a_2=a_3+a_4. This answers the earlier conjecture due to C. J. Smyth and the first author, namely, that there exists no two non-real conjugates of a Pisot number with the same imaginary part and also that at most two conjugates of a Pisot number can have the same real part. On the other hand, we prove that similar four term equations a_1=a_2+a_3+a_4 or a_1+a_2+a_3+a_4 =0 cannot be solved in conjugates of a Pisot number a.

We also show that the roots of the Siegel's polynomial x^3-x-1 are the only solutions to the three term equation a_1+a_2+a_3=0 in conjugates of a Pisot number. Finally, we prove that there exists no Pisot number whose conjugates satisfy the relation a_1=a_2+a_3.

Location

MC - Mathematics & Computer Building

5136B

200 University Avenue West

Waterloo, ON N2L 3G1

Canada

200 University Avenue West

Waterloo, ON N2L 3G1

Canada

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 Indigenous Initiatives Office.