Solution of the conjecture of Schinzel and Zassenhaus and some applications

Abstract: We will detail the full proof of an explicit form of the Schinzel-Zassenhaus conjecture: an algebraic integer of degree $n > 1$ is either a root of unity or else has at least one conjugate of modulus exceeding $2^{1/(4n)}$. We furthermore obtain an extension of the original conjecture over to the setting of holonomic functions, with an application to the smallest critical value for (certain) rational functions.

In another application, we would like to take the occasion to raise the apparently unsolved problem of the essential irreducibility (up-to the cyclotomic factor $X^2-X+1$ in degrees a multiple of 12) of $X^{2g} - X^g(1+X+1/X) + 1$, the characteristic polynomial of the integer reciprocal Perron-Frobenius matrix of the smallest spectral radius in each given dimension. Our explicit Schinzel-Zassenhaus bound allows for at most $10$ factors of each of these polynomials.

number theory

Audience: researchers in the topic

( slides | video )

Warsaw Number Theory Seminar