Number Theory Seminar

Jonas Jankauskas, Department of Pure Mathematics, University of Waterloo

“Binary words, winding numbers and polynomials with interlaced roots.”

Around 1945, R. Salem showed that Pisot numbers can be used to generate sequences of Salem numbers (in terms of their minimal polynomials). In 1977, D. Boyd proved that this construction produces all Salem numbers. Moreover, under special conditions this construction can be inverted: one can reconstruct back Pisot number from two Salem numbers provided that their conjugates on the unit circle interlace. In 2012, J. McKee and C. Smyth described 3 types of interlacing that produce Pisot numbers from pairs of Salem numbers or roots of unity.

In my talk, I give a combinatorial version of the McKee-Smyth theorem via a more general zeros- and-poles counting formula based on reduced length of binary words representing the interlacing pattern of zeros of real and imaginary parts of a meromorphic function on the domain boundary.

Time permitting, I will discuss the connections to stability and control theory and possible further number-theoretical applications of this result.

MC 5479