Gennady Bachman

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20102026

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math.NT2026

Gaps of Binary Numerical Semigroups and of Binary Inclusion-Exclusion Polynomials

Gennady Bachman

Let $p$ be a given modulus, let $u$ be prime to $p$ , and consider the linear permutation $u \cdot n (mod p)$ of the residue system modulo $p$ . Writing $⟨ x ⟩_{p}$ to denote…

math.NT2026

On Heights and Diameters of Ternary Cyclotomic and Inclusion-Exclusion Polynomials

Gennady Bachman

For the $n$ th cyclotomic polynomial $Φ_{n}$ , let $A (n)$ denote the greatest absolute value of its coefficients, its height, and let $D (n)$ denote the difference between its largest a…

math.NT2023

A Note on Heights of Cyclotomic Polynomials

Gennady Bachman, Christopher Bao, Shenlone Wu

We show that for any positive integer $h$ , either $h$ or $h + 1$ is a height of some cyclotomic polynomial $Φ_{n}$ , where $n$ is a product of three distinct primes.

math.NT2021

Coefficients of Unitary Cyclotomic Polynomials of Order Three

Gennady Bachman

A unitary cyclotomic polynomial of order three is a polynomial of the form \[ Φ^*_{PQR}(x)=\frac{(x^{PQR}-1)(x^P-1)(x^Q-1)(x^R-1)}{(x^{PQ}-1)(x^{QR}-1)(x^{RP}-1)(x-1)}, \] where $P…

math.NT2010

On a Class of Ternary Inclusion-Exclusion Polynomials

Gennady Bachman, Pieter Moree

A ternary inclusion-exclusion polynomial is a polynomial of the form \[ Q_{p,q,r}=\frac{(z^{pqr}-1)(z^p-1)(z^q-1)(z^r-1)} {(z^{pq}-1)(z^{qr}-1)(z^{rp}-1)(z-1)}, \] where $p$ , $q$ ,…

math.NT2010

On Ternary Inclusion-Exclusion Polynomials

Gennady Bachman

Taking a combinatorial point of view on cyclotomic polynomials leads to a larger class of polynomials we shall call the inclusion-exclusion polynomials. This gives a more appropria…