Square-free Polynomial Research Articles

Computing abelian varieties over finite fields isogenous to a power

In this paper we give a module-theoretic description of the isomorphism classes of abelian varieties A isogenous to B^r, where the characteristic polynomial g of Frobenius of B is an ordinary square-free q-Weil polynomial, for a power q of a prime p, or a square-free p-Weil polynomial with no real roots. Under some extra assumptions on the polynomial g we give an explicit description of all the isomorphism classes which can be computed in terms of fractional ideals of an order in a finite product of number fields. In the ordinary case, we also give a module-theoretic description of the polarizations of A.

Критерий периодичности непрерывных дробей ключевых элементов гиперэллиптических полей

Периодичность и квазипериодичность функциональных непрерывных дробей в гиперэллиптическом поле $$L = \mathbb{Q}(x)(\sqrt{f})$$ имеет более сложную природу, чем периодичность числовых непрерывных дробей элементов квадратичных полей. Известно, что периодичность непрерывной дроби элемента $$\sqrt{f}/h^{g+1}$$, построенной по нормированию, связанному с многочленом h первой степени, эквивалентна наличию нетривиальных S-единиц в поле L рода g и эквивалентна наличию нетривиального кручения в группе классов дивизоров. В данной статье найден точный промежуток значений $$s \in \mathbb{Z}$$ таких, что элементы $$\sqrt{f}/h^s$$ имеют периодическое разложение в непрерывную дробь, где $$f \in \mathbb{Q}[x]$$ - свободный от квадратов многочлен четной степени. Для многочленов f нечетной степени проблема периодичности непрерывных дробей элементов вида $$\sqrt{f}/h^s$$ рассмотрена в статье [5], причем там доказано, что длина квазипериода не превосходит степени фундаментальной S-единицы поля L. Проблема периодичности непрерывных дробей элементов вида $$\sqrt{f}/h^s$$ для многочленов f четной степени является более сложной. Это подчеркивается найденным нами примером многочлена f степени 4, для которого соответствующие непрерывные дроби имеют аномально большую длину периода. Ранее в статье [5] также были найдены примеры непрерывных дробей элементов гиперэллиптического поля L с длиной квазипериода значительно превосходившей степень фундаментальной S-единицы поля L.

A Verified Implementation of the Berlekamp\u2013Zassenhaus Factorization Algorithm

We formally verify the Berlekamp–Zassenhaus algorithm for factoring square-free integer polynomials in Isabelle/HOL. We further adapt an existing formalization of Yun’s square-free factorization algorithm to integer polynomials, and thus provide an efficient and certified factorization algorithm for arbitrary univariate polynomials. The algorithm first performs factorization in the prime field mathrm {GF}(p){} and then performs computations in the ring of integers modulo p^k, where both p and k are determined at runtime. Since a natural modeling of these structures via dependent types is not possible in Isabelle/HOL, we formalize the whole algorithm using locales and local type definitions. Through experiments we verify that our algorithm factors polynomials of degree up to 500 within seconds.

On ideals generated by two generic quadratic forms in the exterior algebra

Based on the structure theory of pairs of skew-symmetric matrices, we give a conjecture for the Hilbert series of the exterior algebra modulo the ideal generated by two generic quadratic forms. We show that the conjectured series is an upper bound in the coefficient-wise sense, and we determine a majority of the coefficients. We also conjecture that the series is equal to the series of the squarefree polynomial ring modulo the ideal generated by the squares of two generic linear forms.

Liminal reciprocity and factorization statistics

Let M d,n (q) denote the number of monic irreducible polynomials in 𝔽 q [x 1 ,x 2 ,...,x n ] of degree d. We show that for a fixed degree d, the sequence M d,n (q) converges coefficientwise to an explicitly determined rational function M d,∞ (q). The limit M d,∞ (q) is related to the classic necklace polynomial M d,1 (q) by an involutive functional equation we call liminal reciprocity. The limiting first moments of factorization statistics for squarefree polynomials are expressed in terms of symmetric group characters as a consequence of liminal reciprocity, giving a liminal analog of a result of Church, Ellenberg, and Farb.

DENSITY OF POWER‐FREE VALUES OF POLYNOMIALS

We establish asymptotic formulae for the number of $k$-free values of square-free polynomials $F(x_{1},\ldots ,x_{n})\in \mathbb{Z}[x_{1},\ldots ,x_{n}]$ of degree $d\geqslant 2$ for any $n\geqslant 1$, including when the variables are prime, as long as $k\geqslant (3d+1)/4$. This generalizes a work of Browning.

Polynomial Factorization Statistics and Point Configurations in ℝ3

Abstract We use combinatorial methods to relate the expected values of polynomial factorization statistics over $\mathbb{F}_q$ to the cohomology of ordered configurations in $\mathbb{R}^3$ as a representation of the symmetric group. Our method gives a new proof of the twisted Grothendieck–Lefschetz formula for squarefree polynomial factorization statistics of Church, Ellenberg, and Farb.

COMPLEX MOMENTS AND THE DISTRIBUTION OF VALUES OF OVER FUNCTION FIELDS WITH APPLICATIONS TO CLASS NUMBERS

In this paper we investigate the moments and the distribution of $L(1,\chi_D)$, where $\chi_D$ varies over quadratic characters associated to square-free polynomials $D$ of degree $n$ over $\mathbb{F}_q$, as $n\to\infty$. Our first result gives asymptotic formulas for the complex moments of $L(1,\chi_D)$ in a large uniform range. Previously, only the first moment has been computed due to work of Andrade and Jung. Using our asymptotic formulas together with the saddle-point method, we show that the distribution function of $L(1,\chi_D)$ is very close to that of a corresponding probabilistic model. In particular, we uncover an interesting feature in the distribution of large (and small) values of $L(1, \chi_D)$, that is not present in the number field setting. We also obtain $\Omega$-results for the extreme values of $L(1,\chi_D)$, which we conjecture to be best possible. Specializing $n=2g+1$ and making use of one case of Artin's class number formula, we obtain similar results for the class number $h_D$ associated to $\mathbb{F}_q(T)[\sqrt{D}]$. Similarly, specializing to $n=2g+2$ we can appeal to the second case of Artin's class number formula and deduce analogous results for $h_DR_D$ where $R_D$ is the regulator of $\mathbb{F}_q(T)[\sqrt{D}]$.

Stability for hyperplane complements of type B/C and statistics on squarefree polynomials over finite fields

AbstractIn this paper, we explore a relationship between the topology of the complex hyperplane complements ℳBCn(ℂ) in type B/C and the combinatorics of certain spaces of degree-n polynomials over a finite field Fq. This relationship is a consequence of the Grothendieck trace formula and work of Lehrer and Kim. We use it to prove a correspondence between a representation-theoretic convergence result on the cohomology algebras H*(ℳBCn(ℂ);ℂ), and an asymptotic stability result for certain polynomial statistics on monic squarefree polynomials over Fq with non-zero constant term. This result is the type B/C analogue of a theorem due to Church, Ellenberg, and Farb in type A, and we include a new proof of their theorem. To establish these convergence results, we realize the sequences of cohomology algebras of the hyperplane complements as FIW-algebras finitely generated inFIW-degree 2, and we investigate the asymptotic behaviour of general families of algebras with this structure. We prove a negative result implying that this structure alone is not sufficient to prove the necessary convergence conditions. Our proof of convergence for the cohomology algebras involves the combinatorics of their relators.

On the Finiteness of Hyperelliptic Fields with Special Properties and Periodic Expansion of √f

We prove the finiteness of the set of square-free polynomials f ∈ k[x] of odd degree distinct from 11 considered up to a natural equivalence relation for which the continued fraction expansion of the irrationality $$\sqrt {f\left( x \right)} $$ in k((x)) is periodic and the corresponding hyperelliptic field k(x)(√f) contains an S-unit of degree 11. Moreover, it was proved for k = ℚ that there are no polynomials of odd degree distinct from 9 and 11 satisfying the conditions mentioned above.

Plane curves with minimal discriminant

We give lower bounds for the degree of the discriminant with respect to $y$ of squarefree polynomials $f\in \mathbb {K}[x,y]$ over an algebraically closed field of characteristic zero. Depending on the invariants involved in the lower bound, we give a geometrical characterization of those polynomials having minimal discriminant, and we give an explicit construction of all such polynomials in many cases. In particular, we show that irreducible monic polynomials with minimal discriminant coincide with coordinate polynomials. We obtain analogous partial results for the case of nonmonic or reducible polynomials by studying their $GL_2(\mathbb {K}[x])$-orbit and by establishing some combinatorial constraints on their Newton polygon. Our results suggest some natural extensions of the embedding line theorem of Abhyankar-Moh and of the Nagata-Coolidge problem to the case of unicuspidal curves of $\mathbb {P}^1\times \mathbb {P}^1$.

Zeros of quadratic Dirichlet $L$-functions in the hyperelliptic ensemble

We study the $1$-level density and the pair correlation of zeros of quadratic Dirichlet $L$-functions in function fields, as we average over the ensemble $\mathcal{H}_{2g+1}$ of monic, square-free polynomials with coefficients in $\mathbb{F}_q[x]$. In the case of the $1$-level density, when the Fourier transform of the test function is supported in the restricted interval $(\frac{1}{3},1)$, we compute a secondary term of size $q^{-\frac{4g}{3}}/g$, which is not predicted by the Ratios Conjecture. Moreover, when the support is even more restricted, we obtain several lower order terms. For example, if the Fourier transform is supported in $(\frac{1}{3}, \frac{1}{2})$, we identify another lower order term of size $q^{-\frac{8g}{5}}/g$. We also compute the pair correlation, and as for the $1$-level density, we detect lower order terms under certain restrictions; for example, we see a term of size $q^{-g}/g^2$ when the Fourier transform is supported in $(\frac{1}{4},\frac{1}{2})$. The $1$-level density and the pair correlation allow us to obtain non-vanishing results for $L(\frac12,\chi_D)$, as well as lower bounds for the proportion of simple zeros of this family of $L$-functions.

On the problem of periodicity of continued fractions in hyperelliptic fields

We present new results concerning the problem of periodicity of continued fractions which are expansions of quadratic irrationalities in a field , where is a field of characteristic different from 2, , . Let be a square-free polynomial and suppose that the valuation of the field has two extensions and to the field . We set . A deep connection between the periodicity of continued fractions in the field and the existence of -units made it possible to make great advances in the study of periodic and quasiperiodic elements of the field , and also in problems connected with searching for fundamental -units. Using a new efficient algorithm to search for solutions of the norm equation in the field we manage to find examples of periodic continued fractions of elements of the form , which is a fairly rare phenomenon. For the case of an elliptic field , , we describe all square-free polynomials with a periodic expansion of into a continued fraction in the field . Bibliography: 16 titles.

On common zeros of a pair of quadratic forms over a finite field

Let F be a finite field of characteristic distinct from 2, f and g quadratic forms over F, dim⁡f=dim⁡g=n. A particular case of Chevalley's theorem claims that if n≥5, then f and g have a common zero. We give an algorithm, which establishes whether f and g have a common zero in the case n≤4. The most interesting case is n=4. In particular, we show that if n=4 and det⁡(f+tg) is a squarefree polynomial of degree different from 2, then f and g have a common zero. We investigate the orbits of pairs of 4-dimensional forms (f,g) under the action of the group GL4(F), provided f and g do not have a common zero. In particular, it turns out that for any polynomial p(t) of degree at most 4 up to the above action there exist at most two pairs (f,g) such that det⁡(f+tg)=p(t), and the forms f, g do not have a common zero. The proofs heavily use Brumer's theorem and the Hasse–Minkowski theorem.

The distance to square-free polynomials

In this paper, we consider a variant of Turan's problem on the distance from an integer polynomial in $\mathbb{Z}[x]$ to the nea\-rest irreducible polynomial in $\mathbb{Z}[x]$. We prove that for any polynomial $f \in \mathbb{Z}[x]$, there exist infinitely many square-free polynomials $g\in \mathbb{Z}[x]$ such that $L(f-g) \le 2$, where $L(f-g)$ denotes the sum of the absolute values of the coefficients of $f-g$. On the other hand, we show that this inequality cannot be replaced by $L(f-g) \le 1$. For this, for each integer $d \geq 16$ we construct infinitely many polynomials $f \in \mathbb{Z}[x]$ of degree $d$ such that neither $f$ itself nor any $f(x) \pm x^k$, where $k$ is a non-negative integer, is square-free. Polynomials over prime fields and their distances to square-free polynomials are also considered.

Squarefree polynomials with prescribed coefficients

For nonempty subsets S0,…,Sn−1 of a (large enough) finite field F satisfying|S1|,…,|Sn−1|>2or|S1|,|Sn−1|>n−1, we show that there exist a0∈S0,…,an−1∈Sn−1 such thatTn+an−1Tn−1+…+a1T+a0∈F[T] is a squarefree polynomial.

Minimal determinantal representations of bivariate polynomials

For a square-free bivariate polynomial p of degree n we introduce a simple and fast numerical algorithm for the construction of n×n matrices A, B, and C such that det⁡(A+xB+yC)=p(x,y). This is the minimal size needed to represent a bivariate polynomial of degree n. Combined with a square-free factorization one can now compute n×n matrices for any bivariate polynomial of degree n. The existence of such symmetric matrices was established by Dixon in 1902, but, up to now, no simple numerical construction has been found, even if the matrices can be nonsymmetric. Such representations may be used to efficiently numerically solve a system of two bivariate polynomials of small degree via the eigenvalues of a two-parameter eigenvalue problem. The new representation speeds up the computation considerably.

On the periodicity of continued fractions in elliptic fields

Article [1] raised the question of the finiteness of the number of square-free polynomials f ∈ ℚ[h] of fixed degree for which $$\sqrt f $$ has periodic continued fraction expansion in the field ℚ((h)) and the fields ℚ(h)( $$\sqrt f $$ ) are not isomorphic to one another and to fields of the form ℚ(h) $$\left( {\sqrt {c{h^n} + 1} } \right)$$ , where c ∈ ℚ* and n ∈ ℕ. In this paper, we give a positive answer to this question for an elliptic field ℚ(h)( $$\sqrt f $$ ) in the case deg f = 3.

Polynomial Splitting Measures and Cohomology of the Pure Braid Group

We study for each $n$ a one-parameter family of complex-valued measures on the symmetric group $S_n$, which interpolate the probability of a monic, degree $n$, square-free polynomial in $\mathbb{F}_q[x]$ having a given factorization type. For a fixed factorization type, indexed by a partition $\lambda$ of $n$, the measure is known to be a Laurent polynomial. We express the coefficients of this polynomial in terms of characters associated to $S_n$-subrepresentations of the cohomology of the pure braid group $H^{\bullet}(P_n, \mathbb{Q})$. We deduce that the splitting measures for all parameter values $z= -\frac{1}{m}$ (resp. $z= \frac{1}{m}$), after rescaling, are characters of $S_n$-representations (resp. virtual $S_n$-representations.)

Square-free values of multivariate polynomials over function fields in linear sparse sets

Let [Formula: see text] be a square-free polynomial where [Formula: see text] is a field of [Formula: see text] elements. We view [Formula: see text] as a polynomial in the variable [Formula: see text] with coefficients in the ring [Formula: see text]. We study square-free values of [Formula: see text] in sparse subsets of [Formula: see text] which are given by a linear condition. The motivation for our study is an analogue problem of representing square-free integers by integer polynomials, where it is conjectured that setting aside some simple exceptional cases, a square-free polynomial [Formula: see text] takes infinitely many square-free values. Let [Formula: see text] be co-prime to [Formula: see text], and let [Formula: see text]. A consequence of the main result we show is that if [Formula: see text] is sufficiently large with respect to [Formula: see text] and [Formula: see text], then there exist [Formula: see text] such that [Formula: see text] is square-free. Moreover, as [Formula: see text], the last is true for almost all [Formula: see text]. The main result shows that a similar result holds also for other cases. We then generalize the results to multivariate polynomials.