Let R be a principal ideal domain and let $HR$ denote the Hurwitz series ring over R. We first give a recursive splitting formula for a Hurwitz series whose constant coefficient is a product of two coprime nonunits. This construction leads to irreducibility tests for prime-power constant coefficients, decompositions indexed by the distinct prime divisors of the constant coefficient, and upper and lower bounds for the length of every irreducible factorization. Over a discrete valuation domain, the valuation of any selected coefficient yields a sharper length bound. We then introduce the Hurwitz--Newton polygon and prove a product rule on ranges in which the relevant binomial coefficients are units. Primitive one-edge polygons consequently provide Dumas-type irreducibility criteria. Finally, localization is used to combine independent criteria at several prime elements. Examples over Z, localized polynomial rings, and the Gaussian integers illustrate the results.
Building on the bit-complexity framework of Raghavendra-Weitz and the moment-SOS criteria of Gribling-Polak-Slot, we study the effective use of truncated vanishing identities over Boolean polynomial systems. A complete, constructible identity space gives an augmented moment SDP that can be optimized with exact rational...
We establish the dynamical Mordell--Lang conjecture over the complex numbers for a family of radially ramified polynomial skew products in arbitrary base dimension and fiber rank. We assume that one fixed iterate sends the vertical critical locus into the invariant zero section. Our result covers affine-linear bases an...
We introduce and study the omega invariant of a proper ideal in a Noetherian commutative ring, defined as the number of associated primes of the ideal. Our main objective is to investigate this invariant for monomial ideals and their powers. We characterize associated primes through monomial witnesses and provide an al...
Jacob Miller, M. Nasernejad, Stefan O. Tohaneanu· 1 citation· ⚡1
Let $A$ and $B$ be polynomials with integer coefficients. We study nonzero integer solutions of $$y\mid A(x),\qquad x\mid B(y),$$ using quotient transformations that may change the polynomial pair. We call this process \emph{congruence jumping}. We establish a general reciprocal-polynomial rule governing such jumps, wi...
We give an explicit finite sum for the number of ideals of the Lie algebra of strictly lower triangular $n\times n$ matrices over $\mathbb F_q$, valid for every prime power $q$. The sum runs over integer compositions, with weights expressed using ordinary and Gaussian binomial coefficients. A contraction bijection tran...
Let $p$ be a prime. We prove that every compatible branch datum of degree $p$ over the sphere is realizable by a connected branched cover. The three-point case is constructed in residue characteristic $p$. Henrio's moment theorem supplies the distinct-point moment solutions from which we construct a special primitive t...
