Integer Polynomials Research Articles

Bohr recurrence and density of non-lacunary semigroups of ℕ

A subset R R of integers is a set of Bohr recurrence if every rotation on T d \mathbb {T}^d returns arbitrarily close to zero under some non-zero multiple of R R . We show that the set { k ! 2 m 3 n : k , m , n ∈ N } \{k!\, 2^m3^n\colon k,m,n\in \mathbb {N}\} is a set of Bohr recurrence. This is a particular case of a more general statement about images of such sets under any integer polynomial with zero constant term. We also show that if P P is a real polynomial with at least one non-constant irrational coefficient, then the set { P ( 2 m 3 n ) : m , n ∈ N } \{P(2^m3^n)\colon m,n\in \mathbb {N}\} is dense in T \mathbb {T} , thus providing a joint generalization of two well-known results, one of Furstenberg and one of Weyl.

A Family of Fractal-Based Irreducible Polynomials

Summary We provide some fractal-based irreducibility criteria for integer polynomials. The results are obtained using the self-similar pattern of Pascal’s triangle modulo two and by applying a restatement of the Schönemann–Eisenstein’s irreducibility criterion.

In Memoriam: Peter Benjamin Borwein (1953–2020)

Peter Borwein had wide-ranging mathematical interests, and was a pioneer in computational and experimental investigations in many areas. We fondly recall some of his interests and contributions in a number of topics in analysis, number theory, and other areas, especially work where computation and experimentation played an important role. This includes his work on a number of extremal problems regarding Littlewood polynomials and other families of integer polynomials with restricted coefficients, questions regarding the Mahler measure of an integer polynomial, problems on merit factors and Barker sequences, the Prouhet–Tarry–Escott problem, problems of Pólya and Turán regarding sums of the Liouville function, and questions in combinatorial geometry regarding arrangements of points and lines.

On the Connection Between Irrationality Measures and Polynomial Continued Fractions

AbstractLinear recursions with integer coefficients, such as the recursion that generates the Fibonacci sequence $${F}_{n}={F}_{n-1}+{F}_{n-2}$$ F n = F n - 1 + F n - 2 , have been intensely studied over millennia and yet still hide interesting undiscovered mathematics. Such a recursion was used by Apéry in his proof of the irrationality of $$\zeta \left(3\right)$$ ζ 3 , which was later named the Apéry constant. Apéry’s proof used a specific linear recursion that contained integer polynomials (polynomially recursive) and formed a continued fraction; such formulas are called polynomial continued fractions (PCFs). Similar polynomial recursions can be used to prove the irrationality of other fundamental constants such as $$\pi$$ π and $$e$$ e . More generally, the sequences generated by polynomial recursions form Diophantine approximations, which are ubiquitous in different areas of mathematics such as number theory and combinatorics. However, in general it is not known which polynomial recursions create useful Diophantine approximations and under what conditions they can be used to prove irrationality. Here, we present general conclusions and conjectures about Diophantine approximations created from polynomial recursions. Specifically, we generalize Apéry’s work from his particular choice of PCF to any general PCF, finding the conditions under which a PCF can be used to prove irrationality or to provide an efficient Diophantine approximation. To provide concrete examples, we apply our findings to PCFs found by the Ramanujan Machine algorithms to represent fundamental constants such as $$\pi$$ π , $$e$$ e , $$\zeta \left(3\right)$$ ζ 3 , and the Catalan constant. For each such PCF, we demonstrate the extraction of its convergence rate and efficiency, as well as the bound it provides for the irrationality measure of the fundamental constant. We further propose new conjectures about Diophantine approximations based on PCFs. Looking forward, our findings could motivate a search for a wider theory on sequences created by any linear recursions with integer coefficients. Such results can help the development of systematic algorithms for finding Diophantine approximations of fundamental constants. Consequently, our study may contribute to ongoing efforts to answer open questions such as the proof of the irrationality of the Catalan constant or of values of the Riemann zeta function (e.g., $$\zeta \left(5\right)$$ ζ 5 ).

Schinzel-type bounds for the Mahler measure on lemniscates

We study the generalized Mahler measure on lemniscates, and prove a sharp lower bound for the measure of totally real integer polynomials that includes the classical result of Schinzel expressed in terms of the golden ratio. Moreover, we completely characterize many cases when this lower bound is attained. For example, we explicitly describe all lemniscates and the corresponding quadratic polynomials that achieve our lower bound for the generalized Mahler measure. It turns out that the extremal polynomials attaining the bound must have even degree. The main computational part of this work is related to finding many extremals of degree four and higher, which is a new feature compared to the original Schinzel’s theorem where only quadratic irreducible extremals are possible.

FPGA Based Accelerator for Implementation of Large Integer Polynomials

A 13-bit multiplier is implemented on the Artix- 7 100T FPGA using a divide-and-conquer algorithm. The designis coded in SystemVerilog, leveraging its powerful features for hardware description and synthesis. The divide-and-conquer approach breaks down the multiplication task into smaller sub- tasks, enhancing efficiency and reducing complexity. The FPGA’s high- performance capabilities, particularly on the Artix-7 100T board, make it well-suited for accelerating the computations involved. Additionally, Area Delay Product (ADP) tools are employed to evaluate the algorithm’s efficiency. This project aims to showcase the synergy between algorithmic design, hardware implementation, and FPGA capabilities, emphasizing the versa- tility of the Artix-7 100T FPGA in handling complex arithmetic operations.

Revisiting RSA-polynomial problem and semiprime factorization

This paper focuses on the RSA-polynomial problem, a cryptographic hard problem that has been recently proposed and studied in, along with its various applications. We revisit this problem and conduct a refined analysis to address an ambiguous condition that was previously introduced in the context of RSA-polynomial based semiprime factorization. By deriving an accurate attack condition, we are able to identify weak cases of the RSA-polynomial problem and expand the vulnerable bound. To facilitate this, we propose two optimized factoring attacks that leverage improved lattice-based theorems for solving bivariate integer polynomials of a specific form. The validity and effectiveness of our proposed factoring attacks are verified through both theoretical analysis and experimental results. Additionally, we examine the RSA-polynomial based commitment scheme and identify deficiencies that compromise its reliability. To address the limitations, we propose enhancements to the commitment phase of the scheme.

On the Number of Rational Roots of a Rational Polynomial

We propose an extension of Gauss’s lemma and Schönemann-Eisenstein’s irreducibility criterion to determine the impossibility of decomposing an integer polynomial into several polynomials with rational coefficients. This extension allows us to obtain some limitations on the number of rational roots and on the degrees of the factors of a rational polynomial.

CNC: A lightweight architecture for Binary Ring-LWE based PQC

In lattice-based cryptography, Ring Learning with Errors (RLWE) is a computationally hard cryptographic problem, comprising three basic mechanisms i.e., key generation, encryption, and decryption. Binary Ring Learning with Error (BRLWE), a new variant of RLWE has been proposed recently to reduce the key size and computational complexity compared to previous RLWE-based schemes. Based on this BRLWE scheme, efficient hardware architectures have been obtained in recent works for lightweight applications. The key operation involved in this scheme is AB+C , where A and C are integer polynomials and B is a binary polynomial. This paper proposes an efficient hardware architecture for BRLWE-based scheme targeted for lightweight applications. The architecture computes the arithmetic operation AB+C, which includes polynomial multiplication and addition over the polynomial ring Zq/(xn+1). The proposed architecture is applied in two conditions, fixed and variable values of q. Experimental results show the architecture proposed has 50% less Area-Delay Product (ADP) and 20% less Power-Delay Product (PDP) compared to the recently reported work for n=256.

On the divisibility of 7-elongated plane partition diamonds by powers of 8

In 2021 da Silva, Hirschhorn, and Sellers studied a wide variety of congruences for the [Formula: see text]-elongated plane partition function [Formula: see text] by various primes. They also conjectured the existence of an infinite congruence family modulo arbitrarily high powers of 2 for the function [Formula: see text]. We prove that such a congruence family exists — indeed, for powers of 8. The proof utilizes only classical methods, i.e. integer polynomial manipulations in a single function, in contrast to all other known infinite congruence families for [Formula: see text] which require more modern methods to prove.

Finitely generated metabelian groups arising from integer polynomials

Abstract It is shown that there is a finitely generated metabelian group of finite torsion-free rank associated with each non-constant integer polynomial. It is shown how many structural properties of the group can be detected by inspecting the polynomial.

Metric theory of diophantine approximation and asymptotic estimates for the number of polynomials with given discriminants divisible by a large power of a prime number

Discriminants of polynomials characterize the distribution of roots of polynomials in the complex plane. In recent years, for integer polynomials, exact lower-bound estimates have been obtained for the number of polynomials of a given degree and height. The method of obtaining these estimates is based on Minkowski’s theorems in the geometry of numbers and the metric theory of Diophantine approximation. A new method is proposed and allows one to obtain upperbound estimates for the number of polynomials with bounded discriminants in Archimedean and non-Archimedean metrics. The method generalizes the ideas of H. Davenport, B. Volkman, and V. Sprindzuk that allowed them to obtain significant advances in solving Mahler’s problem.

Polynomial orbits in totally minimal systems

Inspired by the recent work of Glasner, Huang, Shao, Weiss and Ye [14], we prove that the maximal ∞-step pro-nilfactor X∞ of a minimal system (X,T) is the topological characteristic factor along polynomials in a certain sense. Namely, we show that by an almost one to one modification of π:X→X∞, the induced open extension π⁎:X⁎→X∞⁎ has the following property: for any d∈N, any open subsets V0,V1,…,Vd of X⁎ with ⋂i=0dπ⁎(Vi)≠∅ and any distinct non-constant integer polynomials pi with pi(0)=0 for i=1,…,d, there exists some n∈Z such that V0∩T−p1(n)V1∩…∩T−pd(n)Vd≠∅.As an application, the following result is obtained: for a totally minimal system (X,T) and integer polynomials p1,…,pd, if every non-trivial integer combination of p1,…,pd is not constant, then there is a dense Gδ subset Ω of X such that the set{(Tp1(n)x,…,Tpd(n)x):n∈Z} is dense in Xd for every x∈Ω.

Hard to Detect Factors of Univariate Integer Polynomials

We investigate the computational complexity of deciding whether a given univariate integer polynomial p(x) has a factor q(x) satisfying specific additional constraints. When the only constraint imposed on q(x) is to have a degree smaller than the degree of p(x) and greater than zero, the problem is equivalent to testing the irreducibility of p(x) and then it is solvable in polynomial time. We prove that deciding whether a given monic univariate integer polynomial has factors satisfying additional properties is NP-complete in the strong sense. In particular, given any constant value k∈Z, we prove that it is NP-complete in the strong sense to detect the existence of a factor that returns a prescribed value when evaluated at x=k (Theorem 1) or to detect the existence of a pair of factors—whose product is equal to the original polynomial—that return the same value when evaluated at x=k (Theorem 2). The list of all the properties we have investigated in this paper is reported at the end of Section Introduction.

HEaaN.MLIR: An Optimizing Compiler for Fast Ring-Based Homomorphic Encryption

Homomorphic encryption (HE) is an encryption scheme that provides arithmetic operations on the encrypted data without doing decryption. For Ring-based HE, an encryption scheme that uses arithmetic operations on a polynomial ring as building blocks, performance improvement of unit HE operations has been achieved by two kinds of efforts. The first one is through accelerating the building blocks, polynomial operations. However, it does not facilitate optimizations across polynomial operations such as fusing two polynomial operations. The second one is implementing highly optimized HE operations in an amalgamated manner. The written codes have superior performance, but they are hard to maintain. To resolve these challenges, we propose HEaaN.MLIR, a compiler that performs optimizations across polynomial operations. Also, we propose Poly and ModArith, compiler intermediate representations (IRs) for integer polynomial arithmetic and modulus arithmetic on integer arrays. HEaaN.MLIR has compiler optimizations that are motivated by manual optimizations that HE developers do. These include optimizing modular arithmetic operations, fusing loops, and vectorizing integer arithmetic instructions. HEaaN.MLIR can parse a program consisting of the Poly and ModArith instructions and generate a high-performance, multithreaded machine code for a CPU. Our experiment shows that the compiled operations outperform heavily optimized open-source and commercial HE libraries by up to 3.06x in a single thread and 4.55x in multiple threads.

Decomposition of multicorrelation sequences and joint ergodicity

AbstractWe show that, under finitely many ergodicity assumptions, any multicorrelation sequence defined by invertible measure-preserving $\mathbb {Z}^d$ -actions with multivariable integer polynomial iterates is the sum of a nilsequence and a nullsequence, extending a recent result of the second author. To this end, we develop a new seminorm bound estimate for multiple averages by improving the results in a previous work of the first, third, and fourth authors. We also use this approach to obtain new criteria for joint ergodicity of multiple averages with multivariable polynomial iterates on ${\mathbb Z}^{d}$ -systems.

ON NEAR-PERFECT NUMBERS OF SPECIAL FORMS

AbstractWe discuss near-perfect numbers of various forms. In particular, we study the existence of near-perfect numbers in the Fibonacci and Lucas sequences, near-perfect values taken by integer polynomials and repdigit near-perfect numbers.

Integrality in the Matching-Jack conjecture and the Farahat-Higman algebra

Using Jack polynomials, Goulden and Jackson have introduced a one parameter deformationτb\tau _bof the generating series of bipartite maps, which generalizes the partition function ofβ\beta-ensembles of random matrices. The Matching-Jack conjecture suggests that the coefficientscμ,νλc^\lambda _{\mu ,\nu }of the functionτb\tau _bin the power-sum basis are non-negative integer polynomials in the deformation parameterbb. Dołęga and Féray have proved in 2016 the “polynomiality” part in the Matching-Jack conjecture, namely that coefficientscμ,νλc^\lambda _{\mu ,\nu }are inQ[b]\mathbb {Q}[b]. In this paper, we prove the “integrality” part, i.e. that the coefficientscμ,νλc^\lambda _{\mu ,\nu }are inZ[b]\mathbb {Z}[b].The proof is based on a recent work of the author that deduces the Matching-Jack conjecture for marginal sumsc¯μ,lλ\overline { c}^\lambda _{\mu ,l}from an analog result for thebb-conjecture, established in 2020 by Chapuy and Dołęga. A key step in the proof involves a new connection with the graded Farahat-Higman algebra.

Hardware-Implemented Lightweight Accelerator for Large Integer Polynomial Multiplication

Large integer polynomial multiplication is frequently used as a key component in post-quantum cryptography (PQC) algorithms. Following the trend that efficient hardware implementation for PQC is emphasized, in this paper, we propose a new hardware-implemented lightweight accelerator for the large integer polynomial multiplication of Saber (one of the National Institute of Standards and Technology third-round finalists). First, we provided a derivation process to obtain the algorithm for the targeted polynomial multiplication. Then, the proposed algorithm is mapped into an optimized hardware accelerator. Finally, we demonstrated the efficiency of the proposed design, e.g., this accelerator with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$v=32$</tex-math></inline-formula> has at least 48.37% less area-delay product (ADP) than the existing designs. The outcome of this work is expected to provide useful references for efficient implementation of other PQC.

Covering systems with large moduli associated with reducible shifts of integer polynomials

A variation of Turán’s polynomial conjecture is studied. Various connections between specific covering systems of congruences and reducible shifts of integer polynomials are established. These results are inspired by related work of A. Schinzel. As applic