Square-free Polynomial Research Articles

Monodromy of stratified braid groups, II

The space of monic squarefree polynomials has a stratification according to the multiplicities of the critical points, called the equicritical stratification. Tracking the positions of roots and critical points, there is a map from the fundamental group of a stratum into a braid group. We give a complete determination of this map. It turns out to be characterized by the geometry of the translation surface structure on CP1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb{C}\\mathbb{P}^1$$\\end{document} induced by the logarithmic derivative df/f\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{d}f/f$$\\end{document} of a polynomial in the stratum.

Exceptional biases in counting primes over function fields

AbstractWe study how often exceptional configurations of irreducible polynomials over finite fields occur in the context of prime number races and Chebyshev's bias. In particular, we show that three types of biases, which we call “complete bias,” “lower order bias,” and “reversed bias,” occur with probability going to zero among the family of all squarefree monic polynomials of a given degree in as , a power of a fixed prime, goes to infinity. The bounds given extend a previous result of Kowalski, who studied a similar question along particular one‐parameter families of reducible polynomials. The tools used are the large sieve for Frobenius developed by Kowalski, an improvement of it due to Perret–Gentil and considerations from the theory of linear recurrence sequences and arithmetic geometry.

Short interval results for powerfree polynomials over finite fields

Let [Formula: see text] be an integer and [Formula: see text] be a finite field with [Formula: see text] elements. We prove several results on the distribution in short intervals of polynomials in [Formula: see text] that are not divisible by the [Formula: see text]th power of any non-constant polynomial. Our main result generalizes a recent theorem by Carmon and Entin on the distribution of squarefree polynomials to all [Formula: see text]. We also develop polynomial versions of the classical techniques used to study gaps between [Formula: see text]-free integers in [Formula: see text]. We apply these techniques to obtain analogs in [Formula: see text] of some classical theorems on the distribution of [Formula: see text]-free integers. The latter results complement the main theorem in the case when the degrees of the polynomials are of moderate size.

Negative Moments of L–Functions With Small Shifts Over Function Fields

Abstract We consider negative moments of quadratic Dirichlet $L$–functions over function fields. Summing over monic square-free polynomials of degree $2g+1$ in $\mathbb{F}_{q}[x]$, we obtain an asymptotic formula for the $k^{\textrm{th}}$ shifted negative moment of $L(1/2+\beta ,\chi _{D})$, in certain ranges of $\beta $ (e.g., when roughly $\beta \gg \log g/g $ and $k<1$). We also obtain non-trivial upper bounds for the $k^{\textrm{th}}$ shifted negative moment when $\log (1/\beta ) \ll \log g$. Previously, almost sharp upper bounds were obtained in [ 3] in the range $\beta \gg g^{-\frac{1}{2k}+\epsilon }$.

The Ratios Conjecture and upper bounds for negative moments of 𝐿-functions over function fields

We prove special cases of the Ratios Conjecture for the family of quadratic Dirichlet L L -functions over function fields. More specifically, we study the average of L ( 1 / 2 + α , χ D ) / L ( 1 / 2 + β , χ D ) L(1/2+\alpha ,\chi _D)/L(1/2+\beta ,\chi _D) , when D D varies over monic, square-free polynomials of degree 2 g + 1 2g+1 over F q [ x ] \mathbb {F}_q[x] , as g → ∞ g \to \infty , and we obtain an asymptotic formula when ℜ β ≫ g − 1 / 2 + ε \Re \beta \gg g^{-1/2+\varepsilon } . We also study averages of products of 2 2 over 2 2 and 3 3 over 3 3 L L -functions, and obtain asymptotic formulas when the shifts in the denominator have real part bigger than g − 1 / 4 + ε g^{-1/4+\varepsilon } and g − 1 / 6 + ε g^{-1/6+\varepsilon } respectively. The main ingredient in the proof is obtaining upper bounds for negative moments of L L -functions. The upper bounds we obtain are expected to be almost sharp in the ranges described above.

Topological techniques in model selection

The LASSO is an attractive regularisation method for linear regression that combines variable selection with an efficient computation procedure. This paper is concerned with enhancing the performance of LASSO for square-free hierarchical polynomial models when combining validation error with a measure of model complexity. The measure of the complexity is the sum of Betti numbers of the model which is seen as a simplicial complex, and we describe the model in terms of components and cycles, borrowing from recent developments in computational topology. We study and propose an algorithm which combines statistical and topological criteria. This compound criterion would allow us to deal with model selection problems in polynomial regression models containing higher-order interactions. Simulation results demonstrate that the compound criteria produce sparser models with lower prediction errors than the estimators of several other statistical methods for higher order interaction models.

Square-free smooth polynomials in residue classes and generators of irreducible polynomials

Building upon the work of A. Booker and C. Pomerance [Proc. Amer. Math. Soc. 145 (2017), pp. 5035–5042], we prove that for a prime power q ≥ 7 q \geq 7 , every residue class modulo an irreducible polynomial F ∈ F q [ X ] F \in \mathbb {F}_q[X] has a non-constant, square-free representative which has no irreducible factors of degree exceeding deg ⁡ F − 1 \deg F -1 . We also give applications to generating sequences of irreducible polynomials.

Energy bounds, bilinear forms and their applications in function fields

We obtain function field analogues of recent energy bounds on modular square roots and modular inversions and apply them to bounding some bilinear forms and to some questions regarding smooth and square-free polynomials in residue classes.

Explicit minimal embedded resolutions of divisors on models of the projective line

Let K be a discretely valued field with ring of integers \(\mathcal {O}_K\) with perfect residue field. Let K(x) be the rational function field in one variable. Let \({\mathbb {P}}^1_{\mathcal {O}_K}\) be the standard smooth model of \({\mathbb {P}}^1_K\) with coordinate x. Let \(f(x) \in \mathcal {O}_K[x]\) be a squarefree polynomial with corresponding divisor of zeroes \({{\,\mathrm{div}\,}}_0(f)\) on \({\mathbb {P}}^1_{\mathcal {O}_K}\). We give an explicit description of the minimal embedded resolution \(\mathcal {Y}\) of the pair \(({\mathbb {P}}^1_{\mathcal {O}_K}, {{\,\mathrm{div}\,}}_0(f))\) by using Mac Lane’s theory to write down the discrete valuations on K(x) corresponding to the irreducible components of the special fiber of \(\mathcal {Y}\).

Toward super‐approximation in positive characteristic

In this note we show that the family of Cayley graphs of a finitely generated subgroup of ${\rm GL}_{n_0}(\mathbb{F}_p(t))$ modulo some admissible square-free polynomials is a family of expanders under certain algebraic conditions. Here is a more precise formulation of our main result. For a positive integer $c_0$, we say a square-free polynomial is $c_0$-admissible if degree of irreducible factors of $f$ are distinct integers with prime factors at least $c_0$. Suppose $\Omega$ is a finite symmetric subset of ${\rm GL}_{n_0}(\mathbb{F}_p(t))$, where $p$ is a prime more than $5$. Let $\Gamma$ be the group generated by $\Omega$. Suppose the Zariski-closure of $\Gamma$ is connected, simply-connected, and absolutely almost simple; further assume that the field generated by the traces of ${\rm Ad}(\Gamma)$ is $\mathbb{F}_p(t)$. Then for some positive integer $c_0$ the family of Cayley graphs ${\rm Cay}(\pi_{f(x)}(\Gamma),\pi_{f(x)}(\Omega))$ as $f$ ranges in the set of $c_0$-admissible polynomials is a family of expanders, where $\pi_{f(t)}$ is the quotient map for the congruence modulo $f(t)$.

Bounds for Polynomials on Algebraic Numbers and Application to Curve Topology

Let $$P \in {\mathbb {Z}}[X, Y]$$ be a given square-free polynomial of total degree d with integer coefficients of bitsize less than $$\tau $$ , and let $$V_{{\mathbb {R}}} (P) := \{ (x,y) \in {\mathbb {R}}^2\mid P (x,y) = 0 \}$$ be the real planar algebraic curve implicitly defined as the vanishing set of P. We give a deterministic algorithm to compute the topology of $$V_{{\mathbb {R}}} (P)$$ in terms of a simple straight-line planar graph $${\mathcal {G}}$$ that is isotopic to $$V_{{\mathbb {R}}} (P)$$ . The upper bound on the bit complexity of our algorithm is in $${\tilde{O}}(d^5 \tau + d^6)$$ (The expression “the complexity is in $${\tilde{O}}(f(d,\tau ))$$ ” with f a polynomial in $$d,\tau $$ is an abbreviation for the expression “there exists a positive integer c such that the complexity is in $$O(({\log d }\log \tau )^c f(d,\tau ))$$ ”); which matches the current record bound for the problem of computing the topology of a planar algebraic curve. However, compared to existing algorithms with comparable complexity, our method does not consider any change of coordinates, and more importantly the returned simple planar graph $${\mathcal {G}}$$ yields the cylindrical algebraic decomposition information of the curve in the original coordinates. Our result is based on two main ingredients: First, we derive amortized quantitative bounds on the roots of polynomials with algebraic coefficients as well as adaptive methods for computing the roots of bivariate polynomial systems that actually exploit this amortization. The results we obtain are more general that the previous literature. Our second ingredient is a novel approach for the computation of the local topology of the curve in a neighborhood of all singular points.

Generalized Hamming weights of toric codes over hypersimplices and squarefree affine evaluation codes

Let $ \mathbb{F}_{q} $ be a finite field with $ q $ elements, where $ q $ is a power of a prime $ p $. A polynomial over $ \mathbb{F}_{q} $ is monomially squarefree if all its monomials are squarefree. In this paper, we determine an upper bound on the number of common zeroes of any set of $ r $ linearly independent monomially squarefree polynomials of $ \mathbb{F}_{q}[t_{1}, t_{2}, \dots, t_{s}] $ in the affine torus $ T = (\mathbb{F}_{q}^{*})^{s} $ under certain conditions on $ r $, $ s $ and the degree of these polynomials. Applying the results, we obtain the generalized Hamming weights of toric codes over hypersimplices and squarefree affine evaluation codes, as defined in [14].

In Brief

This article introduces several efficient and easy-to-use tools to analyze the intersection curve between two quadrics, on the basis of the study of its projection on a plane (the so-called cutcurve) to perform the corresponding lifting correctly. This approach is based on an efficient way of determining the topology of the cutcurve through only solving one degree eight (at most) univariate equation and several quadratic univariate equations, intersecting two pairs of conics and, when the parameterization of the cutcurve in closed form cannot be determined, computing the real roots of several degree four univariate squarefree polynomials whose number (of real roots) is known in advance.

The moments and statistical distribution of class number of primes over function fields

We investigate the moment and the distribution of L(1,χP), where χP varies over quadratic characters associated to irreducible polynomials P of degree 2g+1 over Fq[T] as g→∞. In the first part of the paper, we compute the integral moments of the class number hP associated to quadratic function fields with prime discriminants P, and this is done by adapting to the function field setting some of the previous results carried out by Nagoshi in the number field setting. In the second part of the paper, we compute the complex moments of L(1,χP) in large uniform range and investigate the statistical distribution of the class numbers by introducing a certain random Euler product. The second part of the paper is based on recent results carried out by Lumley when dealing with square-free polynomials.

Tools for analyzing the intersection curve between two quadrics through projection and lifting

This article introduces several efficient and easy-to-use tools to analyze the intersection curve between two quadrics, on the basis of the study of its projection on a plane (the so-called cutcurve) to perform the corresponding lifting correctly. This approach is based on an efficient way of determining the topology of the cutcurve through only solving one degree eight (at most) univariate equation and several quadratic univariate equations, intersecting two pairs of conics and, when the parameterization of the cutcurve in closed form cannot be determined, computing the real roots of several degree four univariate squarefree polynomials whose number (of real roots) is known in advance.

Representation of a polynomial as the sum of an irreducible polynomial and a square-free polynomial

Let ${\mathbb F }$ be a finite field with $q$ elements. We prove that every polynomial $M\in {\mathbb F }[T]$ of degree large enough is a sum $P+Q$ where $P$ is an irreducible polynomial with $\deg P=\deg M$ and $Q$ is a square-free polynomial with $\deg

Counting points on superelliptic curves in average polynomial time

We describe the practical implementation of an average polynomial-time algorithm for counting points on superelliptic curves defined over $\mathbb Q$ that is substantially faster than previous approaches. Our algorithm takes as input a superelliptic curves $y^m=f(x)$ with $m\ge 2$ and $f\in \mathbb Z[x]$ any squarefree polynomial of degree $d\ge 3$, along with a positive integer $N$. It can compute $\#X(\mathbb F_p)$ for all $p\le N$ not dividing $m\mathrm{lc}(f)\mathrm{disc}(f)$ in time $O(md^3 N\log^3 N\log\log N)$. It achieves this by computing the trace of the Cartier--Manin matrix of reductions of $X$. We can also compute the Cartier--Manin matrix itself, which determines the $p$-rank of the Jacobian of $X$ and the numerator of its zeta function modulo~$p$.

Variation of a Theme of Landau--Shanks in Positive Characteristic

Let $\mathbf{A} := \mathbb{F}_q[t]$ be a polynomial ring over a finite field $\mathbb{F}_q$ of odd characteristic and let $D \in \mathbf{A}$ be a square-free polynomial. Denote by $\mathbf{N}_{D}(n,q)$ the number of polynomials $f$ in $\mathbf{A}$ of degree $n$ which may be represented in the form $u \cdot f = A^2-DB^2$ for some $A,B \in \mathbf{A}$ and $u \in \mathbb{F}_q^{\times}$, and by $\mathbf{B}_{\mathcal{D}}(n,q)$ the number of polynomials in $\mathbf{A}$ of degree $n$ which can be represented by a primitive quadratic form of a given discriminant $\mathcal{D} \in \mathbf{A}$, not necessary square-free. If the class number of the maximal order of $\mathbb{F}_q(t,\sqrt{D})$ is one, then we give very precise asymptotic formulas for $\mathbf{N}_{D}(n,q)$. Moreover, we also give very precise asymptotic formulas for $\mathbf{B}_{\mathcal{D}}(n,q)$.

The distance to a squarefree polynomial over $\mathbb F_2[x

The distance to a squarefree polynomial over $\mathbb F_2[x

On square-free values of large polynomials over the rational function field

AbstractWe investigate the density of square-free values of polynomials with large coefficients over the rational function field 𝔽q[t]. Some interesting questions answered as special cases of our results include the density of square-free polynomials in short intervals, and an asymptotic for the number of representations of a large polynomial N as a sum of a k-th power of a small polynomial and a square-free polynomial.