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Bhawesh Mishra

Publications and source records attributed to Bhawesh Mishra.

13 recordsLinked to original sources
arXiv2026

The Dynamical Mordell--Lang Conjecture for Relatively Étale Systems over Products of Curves

We establish the dynamical Mordell--Lang conjecture over \(\mathbb C\) for endomorphisms that are relatively étale over arbitrary endomorphisms of finite products of smooth projective curves. In particular, we establish the conjecture for arbitrary endomorphisms of product of smooth projective curves and relatively étale skew products. The arithmetic input is our unconditional theorem for split endomorphisms over \(\overline{\mathbb Q}\), proved by reducing to a simultaneous Hasse principle for the local periods of critical points and analyzing local inertia in joint arboreal towers. We combine this arithmetic input with relative étaleness, simultaneous algebraic specialization preserving wandering base coordinates, prescribed-reduction \(p\)-adic embeddings, and \(p\)-adic interpolation to obtain the result in the complex case.

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arXiv2026

Dynamical Mordell--Lang Conjecture for Higher-Rank Radially Ramified Skew Products

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 and, under a degree-gap condition, nonlinear étale polynomial bases. We also establish a uniform Skolem--Mahler--Lech theorem for a fixed nonlinear orbit and a fixed polynomial-coefficient linear recurrence whose trailing coefficient is a nonzero constant. The theorem applies to joint polynomial and rational relations involving the orbit and finitely many successive terms of any solution. A single eventual period works for every such solution and relation, although the finite exceptional set may depend on the relation. Our proof combines controlled nonarchimedean realizations with an orbitwise transfer theorem for superattracting line-bundle dynamics.

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arXiv2026

Simultaneous Periods for Families of Rational Maps Modulo Primes

Let $K$ be a number field, and $φ_{1},\ldots,φ_{g}\in K(t)$ be finitely many rational maps, each of degree at least $2$. We first show that for generic finite sets $\mathcal{A}_{1},\ldots,\mathcal{A}_{g}$ consisting entirely of points that are not $φ_{i}$-periodic, there exists a set of primes $\mathfrak p$ of $K$ of positive density such that for each $\mathcal{A}_{i}$ and every $α\in\mathcal{A}_i$, $α$ is not $φ_i$-periodic modulo $\mathfrak p$. The notion of genericity used here is defined in terms of the associated arboreal fields and is sharper than those previously used in the literature. Leveraging our proof in the generic case, we then show that the same conclusion holds for most \textit{expected} cases of non-generic sets $\mathcal{A}_{i}$. Finally, we apply our result to confirm the dynamical Mordell--Lang conjecture for coordinate-wise actions of a class of maps that includes rational maps that are generic in this sense.

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arXiv2025

Polynomials with factors of the form $(x^q-a)$ with roots modulo every integer

Given an odd prime $q$, a natural number $l$ and non-zero $q$-free integers $a_{1}, a_{2}, \ldots, a_{l}$, none of which are equal to $1$ or $-1$, we give necessary and sufficient conditions for the polynomial $\prod_{j=1}^{l} (x^{q} - a_{j})$ to have roots modulo every positive integer. Consequently: (i) if $l \leq q$ and none of $a_{1}, a_{2}, \ldots, a_{l}$ is a perfect $q^{th}$ power, then the polynomial $\prod_{j=1}^{l} (x^{q} - a_{j})$ fails to have roots modulo some positive integer; $(ii)$ For every $l\in\mathbb{N}$, and every $(c_{j})_{j=1}^{l}\in\big(\mathbb{F}_{q}\setminus\{0\}\big)^{l}$, the polynomial $\prod_{j=1}^{l} (x^{q} - a_{j})$ has roots modulo every positive integer if and only if $\prod_{j=1}^{l} (x^{q} - \text{rad}_{q}\big(a_{j}^{c_{j}}\big)))$ has roots modulo every positive integer. Here $\text{rad}_{q}(a_{j})$ denotes the $q$-free part of the integer $a_{j}$.

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arXiv2025

Blocking Sets and Power Residue Modulo Integers with Bounded Number of Prime Factors

Let $q$ be an odd prime and $k$ be a natural number. We show that a finite subset of integers $S$ that does not contain any perfect $q^{th}$ power, contains a $q^{th}$ power residue modulo almost every natural numbers $N$ with at most $k$ prime factors if and only if $S$ corresponds to a $k$-blocking set of $\PG(\mathbb{F}_{q}^{n})$. Here, $n$ is the number of distinct primes that divides the $q$-free parts of elements of $S$. Consequently, this geometric connection enables us to utilize methods from Galois geometry to derive lower bounds for the cardinalities of such sets $S$ and to completely characterize such $S$ of the smallest and the second smallest cardinalities. Furthermore, the property of whether a finite subset of integers contains a $q^{th}$ power residue modulo almost every integer $N$ with at most $k$ prime factors is invariant under the action of projective general linear group $\mathrm{PGL}(n, q)$.

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arXiv2025

Prime Power Residues and Blocking Sets

Let $q$ be a fixed odd prime. We show that a finite subset $B$ of integers, not containing any perfect $q^{th}$ power, contains a $q^{th}$ power modulo almost every prime if and only if $B$ corresponds to a blocking set (with respect to hyperplanes) in $\mathrm{PG}(\mathbb{F}_{q}^{k})$. Here, $k$ is the number of distinct prime divisors of $q$-free parts of elements of $B$. As a consequence, the property of a subset $B$ to contain $q^{th}$ power modulo almost every prime $p$ is invariant under geometric $q$-equivalence defined by an element of the projective general linear group $\mathrm{PGL}(\mathbb{F}_{q}^{k})$. Employing this connection between two disparate branches of mathematics, Galois geometry and number theory, we classify, and provide bounds on the sizes of, minimal such sets $B$.

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arXiv2024

A Generalization of the Grunwald-Wang Theorem for $n^{th}$ Powers

Let $n$ be a natural number greater than $2$ and $q$ be the smallest prime dividing $n$. We show that a finite subset $A$ of rationals, of cardinality at most $q$, contains a $n^{th}$ power in $\mathbb{Q}_{p}$ for almost every prime $p$ if and only if $A$ contains a perfect $n^{th}$ power, barring some exceptions when $n$ is even. This generalizes the Grunwald-Wang theorem for $n^{th}$ powers, from one rational number to finite subsets of rational numbers. We also show that the upper bound $q$ in this generalization is optimal for every $n$.

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arXiv2023

Almost No Finite Subset of Integers Contains a $q^{th}$ Power Modulo Almost Every Prime

Let $q$ be a prime. We give an elementary proof of the fact that for any $k\in\mathbb{N}$, the proportion of $k$-element subsets of $\mathbb{Z}$ that contain a $q^{th}$ power modulo almost every prime, is zero. This result holds regardless of whether the proportion is measured additively or multiplicatively. More specifically, the number of $k$-element subsets of $[-N, N]\cap\mathbb{Z}$ that contain a $q^{th}$ power modulo almost every prime is no larger than $a_{q,k} N^{k-(1-\frac{1}{q})}$, for some positive constant $a_{q,k}$. Furthermore, the number of $k$-element subsets of $\{\pm p_{1}^{e_{1}} p_{2}^{e_{2}} \cdots p_{N}^{e_{N}} : 0 \leq e_{1}, e_{2}, \ldots, e_{N}\leq N\}$ that contain a $q^{th}$ power modulo almost every prime is no larger than $m_{q,k} \frac{N^{Nk}}{q^{N}}$ for some positive constant $m_{q,k}$.

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arXiv2023

Prime Power Residue and Linear Coverings of Vector Space over $\mathbb{F}_{q}$

Let $q$ be an odd prime and $B = \{b_{j}\}_{j=1}^{l}$ be a finite set of nonzero integers that does not contain a perfect $q^{th}$ power. We show that $B$ has a $q^{th}$ power modulo every prime $p \neq q$ and not dividing $\prod_{b\in B} b$ if and only if $B$ corrresponds to a linear hyperplane covering of $\mathbb{F}_{q}^{k}$. Here, $k$ is the number of distinct prime factors of the $q$-free part of elements of $B$. Consequently: $(i)$ a set $B \subset\mathbb{Z}\setminus\{0\}$ with cardinality less than $q+1$ cannot have a $q^{th}$ power modulo almost every prime unless it contains a perfect $q^{th}$ power and $(ii)$ For every set $B = \{b_{j}\}_{j=1}^{l} \subset\mathbb{Z}\setminus\{0\}$ and for every $\big(c_{j}\big)_{j=1}^{l} \in\Big(\mathbb{F}_{q}\setminus\{0\}\Big)^{l}$ the set $B$ contains a $q^{th}$ power modulo every prime $p \neq q$ and not dividing $\prod_{j=1}^{l}$ if and only if the set $\{b_{j}^{c_{j}}\}_{j=1}^{l}$ does so.

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arXiv2022

Polynomials over Ring of Integers of Global Fields that have Roots Modulo Every Finite Indexed Subgroup

A polynomial with coefficients in the ring of integers $\mathcal{O}_{K}$ of a global field $K$ is called intersective if it has a root modulo every finite-indexed subgroup of $\mathcal{O}_{K}$. We prove two criteria for a polynomial $f(x)\in\mathcal{O}_{K}[x]$ to be intersective. One of these criteria is in terms of the Galois group of the splitting field of the polynomial, whereas the second criterion is verifiable entirely in terms of constants which depend upon $K$ and the polynomial $f$. The proofs use the theory of global field extensions and upper bound on the least prime ideal in the Chebotarev density theorem.

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arXiv2021

Minimally Intersective Polynomials with Arbitrarily Many Quadratic Factors

Given a natural number $n \geq 4$ we show that there exists infinitely many polynomials $f_{n}(x):= \prod_{i=1}^{n} (x^{2} - a_{i})$ such that (i) $f_{n}(x)$ has a root modulo every positive integer, (ii) $f_{n}(x)$ has no rational roots, and (iii) every proper divisor of $f_{n}(x)$ fails to have root modulo some positive integer. We exhibit a process to explicitly construct such $f_{n}$ and this process demonstrates that the set of natural numbers $a_{n}$, such that the polynomial $f_{n}(x):= \prod_{i=1}^{n} (x^{2} - a_{i})$ satisfies the properties (i), (ii) and (iii), is of positive asymptotic density in $\mathbb{N}$.

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arXiv2021

Intersective Polynomials Arising from Sums of Powers

Given a natural number $n \geq 2$, an integer $k$ and for a judiciously chosen $l = l(n)$ we give necessary and sufficient conditions for the polynomial $f_{n,k} = \big( \sum_{i=1}^{l} x_{i}^{n} \big) - k$ to have roots modulo every positive integer.

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