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Gennady Bachman

Publications and source records attributed to Gennady Bachman.

7 recordsLinked to original sources
arXiv2026

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

Let $p$ be a given modulus, let $u$ be prime to $p$, and consider the linear permutation $u\cdot n\pmod p$ of the residue system modulo $p$. Writing $\langle x\rangle_p$ to denote the least nonnegative residue of $x$ modulo $p$, we say that a pair of integers $(a,b)$ is a dominant pair of this permutation if either the inequality $\max(\langle ua\rangle_p,\langle ub\rangle_p)<\min_{a \max_{a<n<b}\langle un\rangle_p$ hold. The main technical part of this work gives analysis of this property of linear permutations of residue systems. We then apply this analysis to the problems that motivated it, and give (i) complete description of the gapsets of binary inclusion-exclusion polynomials $Q_{\{p,q\}}$ (which include binary cyclotomic polynomials $Φ_{pq}$ as its principal special case), and (ii) complete description of all possible distances between consecutive elements of a numerical semigroup $\langle p,q\rangle$.

math.NT↗
arXiv2026

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

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 and smallest coefficients, its diameter. We show that for any odd prime $p$ and an integer $h$ in the range $1\le h\le(p+1)/2$, there are arbitrarily large primes $q$ and $r$ such that $Φ_{pqr}$ has the height $h$. This certainly answers the question of whether every natural number occurs as the height of some cyclotomic polynomial. Our construction specifies explicit choices of $q$ and $r$ with $A(pqr)=h$, and for these choices $D(pqr)$ has one of two values: it is either $2h$ or $2h-1$, depending on the congruence class of $h$ modulo $p$.

math.NT↗
arXiv2025

An Unimaginative Proof of Fermat's Two Squares Theorem

We give a simple direct proof of Fermat's two squares theorem. Our argument uses no intricate notions or ideas; one might say that it is a proof by careful bookkeeping. As such, the proof may be particularly easy to comprehend by students who are just starting out in math.

math.HO↗
arXiv2021

Coefficients of Unitary Cyclotomic Polynomials of Order Three

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 $P$, $Q$ and $R$ are powers of three distinct primes $p$, $q$ and $r$. Fixing any such prime triple generates a family of these polynomials corresponding to all possible choices of $P=p^a$, $Q=q^b$ and $R=r^c$. We study the coefficients of polynomials in such a family. In particular, we show that the coefficients of polynomials in every such family cover all of $\mathbb{Z}$.

math.NT↗
arXiv2010

On Ternary Inclusion-Exclusion Polynomials

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 appropriate setting for certain types of questions about the coefficients of these polynomials. After establishing some basic properties of inclusion-exclusion polynomials we turn to a detailed study of the structure of ternary inclusion-exclusion polynomials. The latter subclass is exemplified by cyclotomic polynomials $Φ_{pqr}$, where $p<q<r$ are odd primes. Our main result is that the set of coefficients of $Φ_{pqr}$ is simply a string of consecutive integers which depends only on the residue class of $r$ modulo $pq$.

math.NT↗
arXiv2010

On a Class of Ternary Inclusion-Exclusion Polynomials

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 $p$, $q$, and $r$ are integers $\ge3$ and relatively prime in pairs. This class of polynomials contains, as its principle subclass, the ternary cyclotomic polynomials corresponding to restricting $p$, $q$, and $r$ to be distinct odd prime numbers. Our object here is to continue the investigation of the relationship between the coefficients of $Q_{p,q,r}$ and $Q_{p,q,s}$, with $r\equiv s\pmod{pq}$. More specifically, we consider the case where $1\le s<\max(p,q)<r$, and obtain a recursive estimate for the function $A(p,q,r)$--the function that gives the maximum of the absolute values of the coefficients of $Q_{p,q,r}$. A simple corollary of our main result is the following absolute estimate. If $s\ge1$ and $r\equiv\pm s\pmod{pq}$, then $A(p,q,r)\le s$.

math.NT↗