Number Theory seminar

Jonas Jankauskas, Pure Mathematics, University of Waterloo

"There are no two non-real conjugates of a Pisot number with the same imaginary part"

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.