Yuri Bilu, Florian Luca
We show that, apart from some obvious exceptions, the number of trinomials vanishing at given complex numbers is bounded by an absolute constant. When the numbers are algebraic, we also bound effectively the degrees and the heights of these trinomials.
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Riffaut (2019) conjectured that a singular modulus of degree $h\ge 3$ cannot be a root of a trinomial with rational coefficients. We show that this conjecture follows from the GRH, and obtain partial unconditional results.
We show that univariate trinomials $x^n + ax^s + b \in \mathbb{F}_q[x]$ can have at most $δ\Big\lfloor \frac{1}{2} +\sqrt{\frac{q-1}δ} \Big\rfloor$ distinct roots in $\mathbb{F}_q$, where $δ= \gcd(n, s, q - 1)$. We also derive explicit trinomials having $\sqrt{q}$ roots in $\mathbb{F}_q$ when $q$ is square and $δ=1$, thus showing that our bound is tight for an infinite family of finite fields and trinomials. Furthermore, we present the results of a large-scale computation which suggest that an $O(δ\log q)$ upper bound may be possible for the special case where $q$ is prime. Finally, we give a …
We give a separation bound for the complex roots of a trinomial $f \in \mathbb{Z}[X]$. The logarithm of the inverse of our separation bound is polynomial in the size of the sparse encoding of $f$; in particular, it is polynomial in $\log (°f)$. It is known that no such bound is possible for 4-nomials (polynomials with 4 monomials). For trinomials, the classical results (which are based on the degree of $f$ rather than the number of monomials) give separation bounds that are exponentially worse.As an algorithmic application, we show that the number of real roots of a trinomial $f$ can be comput…
Given a quintic number field $K/\mathbb{Q}$, we study the set of irreducible trinomials, polynomials of the form $x^{5} + ax + b$, that have a root in $K$. We show that there is a genus four curve $C_{K}$ whose rational points are in bijection with such trinomials. This curve $C_{K}$ maps to an elliptic curve defined over a number field, and using this map, we are able (in some cases) to determine all the rational points on $C_{K}$ using elliptic curve Chabauty.
Let $K$ be a number field generated by a root $þ$ of a monic irreducible trinomial $F(x) = x^n+ax^{m}+b \in \Z[x]$. In this paper, we study the problem of $K$. More precisely, we provide some explicit conditions on $a$, $b$, $n$, and $m$ for which $K$ is not monogenic. As applications, we show that there are infinite families of non-monogenic number fields defined by trinomials of degree $n=2^r\cdot3^k$ with $r$ and $k$ two positive integers. We also give infinite families of non-monogenic sextic number fields defined by trinomials. Some illustrating examples are giving at the end of this pape…
On common index divisors and monogenity of certain number fields defined by trinomials of type $x^{2^r}+ax^m+b$
该研究利用$p$-进牛顿多边形技术与Ore指数定理，分析$x^{2^r}+ax^m+b$型三项式定义数域的单基因性，给出非单基因的显式条件，构造出自身非单基因但根生成单基因数域的三项式族，并辅以计算示例。
Let $K=\Q(θ)$ be a number field generated by a complex root $þ$ of a monic irreducible trinomial $F(x) = x^n+ax+b \in \Z[x]$. There is an extensive literature of monogenity of number fields defined by trinomials, Gaál studied the multi-monogenity of sextic number fields defined by trinomials. Jhorar and Khanduja studied the integral closedness of $\Z[þ]$. But if $ \Z[þ]$ is not integrally closed, then Jhorar and Khanduja's results cannot answer on the monogenity of $K$. In this paper, based on Newton polygon techniques, we deal with the problem of monogenity of $K$. More precisely, when $\Z_K …
On non-monogenic number fields defined by trinomials of type $x^n +ax^m+b$
本文研究了由 $x^n+ax^m+b$ 型三项式定义的数域的非单生性问题，给出了判定非单生成的显式条件，并证明了存在无限族的非单生成数域。
Finite Planes, Zigzag Sequences, Fibonacci Numbers, Artin's Conjecture and Trinomials
本文从初等阿贝尔群的忠实矩阵表示出发，构造出一类满足三项递推且与锯齿零一序列相关的新多项式，并揭示其与斐波那契数、Artin 猜想及有限域上三项式分解的深层联系。
On common index divisors and monogenity of of the nonic number field defined by a trinomial $x^9+ax+b$
该研究针对由三项式$x^9+ax+b$定义的九次数域，给出了素公共指标因子的充要条件，确定了无穷多非单一九次数域参数族，并辅以数值例子验证理论结果。
Irreducible trinomials of given degree n over $F_2$ do not always exist and in the cases that there is no irreducible trinomial of degree n it may be effective to use trinomials with an irreducible factor of degree n. In this paper we consider some conditions under which irreducible polynomials divide trinomials over $F_2$. A condition for divisibility of self-reciprocal trinomials by irreducible polynomials over $F_2$ is established. And we extend Welch's criterion for testing if an irreducible polynomial divides trinomials $x^m+x^s+1$ to the trinomials $x^{am}+x^{bs}+1$.
Involutary pemutations over finite fields given by trinomials and quadrinomials
针对满足特定同余条件的有限域，本文构造了由三项式和四项式给出的对合置换多项式，给出了系数的显式公式，并证明了这些多项式在生成具有特定数量不动点的对合时的数量下界。
Let K be a field and t>=0. Denote by Bm(t,K) the maximum number of non-zero roots in K, counted with multiplicities, of a non-zero polynomial in K[x] with at most t+1 monomial terms. We prove, using an unified approach based on Vandermonde determinants, that Bm(t,L)<=t^2 Bm(t,K) for any local field L with a non-archimedean valuation v such that v(n)=0 for all non-zero integer n and residue field K, and that Bm(t,K)<=(t^2-t+1)(p^f-1) for any finite extension K/Qp with residual class degree f and ramification index e, assuming that p>t+e. For any finite extension K/Qp, for p odd, we also show th…
We obtain upper bounds for the number of monic irreducible polynomials over $\mathbb Z$ of a fixed degree $n$ and a growing height $H$ for which the field generated by one of its roots has a given discriminant. We approach it via counting square-free parts of polynomial discriminants via two complementing approaches. In turn, this leads to a lower bound on the number of distinct discriminants of fields generated by roots of polynomials of degree $n$ and height at most $H$. We also give an upper bound for the number of trinomials of bounded height with given square-free part of the discriminant…
The Motzkin numbers $M_n=\sum_{k=0}^n\binom n{2k}\binom{2k}k/(k+1)$ $(n=0,1,2,\ldots)$ and the central trinomial coefficients $T_n$ ($n=0,1,2,\ldots)$ given by the constant term of $(1+x+x^{-1})^n$, have many combinatorial interpretations. In this paper we establish the following surprising arithmetic properties of them with $n$ any positive integer: $$\frac2n\sum_{k=1}^n(2k+1)M_k^2\in\mathbb Z,$$ $$\frac{n^2(n^2-1)}6\,\bigg|\,\sum_{k=0}^{n-1}k(k+1)(8k+9)T_kT_{k+1},$$ and also $$\sum_{k=0}^{n-1}(k+1)(k+2)(2k+3)M_k^23^{n-1-k}=n(n+1)(n+2)M_nM_{n-1}.$$
On the monogenity of quartic number fields defined by $x^4+ax^2+b$
本文研究了由不可约三项式 $x^4+ax^2+b$ 的根生成的四次数域的单生性，刻画了 $Z[α]$ 整闭的条件，给出了 $p=2,3$ 时整除公因子的最高幂次，并证明了此类域中仅存在唯一的幂整基生成元。
We determine the limit of the rate $\frac{ν_{n,a}}{n}$ between the number $ν_{n,a}$ of roots of the trinomial $x^n-ax-1$, $a\in (0,2]$, which are greater than 1 in modulus, and degree $n$. The analogue of Boyd's Conjecture (C) for Perron numbers is a consequence of the limit, under the assumption that the conjecture of Lind-Boyd is valid. The product of these $ν_{n,a}$ roots has also a limit when $n\to\infty$. The explicit expression of the limit by an integral is presented. The computing of the rate and the product for $n=100,150$ as well as of its limits is presented.
We study the number of points in the family of plane curves defined by a trinomial \[ \mathcal{C}(α,β)= \{(x,y)\in\mathbb{F}_q^2\,:\,αx^{a_{11}}y^{a_{12}}+βx^{a_{21}}y^{a_{22}}=x^{a_{31}}y^{a_{32}}\} \] with fixed exponents (not collinear) and varying coefficients over finite fields. We prove that each of these curves has an almost predictable number of points, given by a closed formula that depends on the coefficients, exponents, and the field, with a small error term $N(α,β)$ that is bounded in absolute value by $2\tilde{g}q^{1/2}$, where $\tilde{g}$ is a constant that depends only on the ex…
The index of the septic number field defined by $x^7+ax^5+b$
本文研究由不可约三项式 $x^7+ax^5+b$ 生成的七次数域，证明其指数只能为 $1$、$2$ 或 $4$，从而回答 Narkiewicz 问题 22，并给出数域非单基因的充分条件。
On the index divisors and monogenity of certain nonic number fields
本文研究由三项式 $x^9+ax+b$ 的根生成的九次数域，刻画了素数整除其指数 $i(K)$ 的条件并给出指数的素数幂分解，从而部分回答了 Narkiewicz 问题 22，并指出指数非 1 时数域非单基因。
We study the number of primes with a given primitive root and in an arithmetic progression under the assumption of a suitable form of the generalized Riemann Hypothesis. Previous work of Lenstra, Moree and Stevenhagen has given asymptotics without an explicit error term, we provide an explicit error term by combining their work with the method of Hooley regarding Artin's primitive root conjecture. We give an application to a Diophantine problem involving primes with a given primitive root.
The first purpose of our paper is to show how Hooley's celebrated method leading to his conditional proof of the Artin conjecture on primitive roots can be combined with the Hardy-Littlewood circle method. We do so by studying the number of representations of an odd integer as a sum of three primes, all of which have prescribed primitive roots. The second purpose is to analyse the singular series. In particular, using results of Lenstra, Stevenhagen and Moree, we provide a partial factorisation as an Euler product and prove that this does not extend to a complete factorisation.
A famous conjecture of Artin states that there are infinitely many prime numbers for which a fixed integer $g$ is a primitive root, provided $g \neq -1$ and $g$ is not a perfect square. Thanks to work of Hooley, we know that this conjecture is true, conditional on the truth of the Generalized Riemann Hypothesis. Using a combination of Hooley's analysis and the techniques of Maynard-Tao used to prove the existence of bounded gaps between primes, Pollack has shown that (conditional on GRH) there are bounded gaps between primes with a prescribed primitive root. In the present article, we provide …
Fix an integer $g \neq -1$ that is not a perfect square. In 1927, Artin conjectured that there are infinitely many primes for which $g$ is a primitive root. Forty years later, Hooley showed that Artin's conjecture follows from the Generalized Riemann Hypothesis (GRH). We inject Hooley's analysis into the Maynard--Tao work on bounded gaps between primes. This leads to the following GRH-conditional result: Fix an integer $m \geq 2$. If $q_1 < q_2 < q_3 < \dots$ is the sequence of primes possessing $g$ as a primitive root, then $\liminf_{n\to\infty} (q_{n+(m-1)}-q_n) \leq C_m$, where $C_m$ is a f…
It is known that the weight (that is, the number of nonzero coefficients) of a univariate polynomial over a field of characteristic zero is larger than the multiplicity of any of its nonzero roots. We extend this result to an appropriate statement in positive characteristic. Furthermore, we present a new proof of the original result, which produces also the exact number of monic polynomials of a given degree for which the bound is attained. A similar argument allows us to determine the number of monic polynomials of a given degree, multiplicity of a given nonzero root, and number of nonzero co…
Let $m$ be a natural number, and let $\mathcal{Q}$ be a set containing at least $\exp(C m)$ primes. We show that one can find infinitely many strings of $m$ consecutive primes each of which has some $q\in\mathcal{Q}$ as a primitive root, all lying in an interval of length $O_{\mathcal{Q}}(\exp(C'm))$. This is a bounded gaps variant of a theorem of Gupta and Ram Murty. We also prove a result on an elliptic analogue of Artin's conjecture. Let $E/\mathbb{Q}$ be an elliptic curve with an irrational $2$-torsion point. Assume GRH. Then for every $m$, there are infinitely many strings of $m$ consecut…
Given an elliptic curve over a field of characteristic different from 2,3, its discriminant defines a $μ_{12}$-torsor over the field. In this paper, we give an explicit description of this $μ_{12}$-torsor in terms of the 3-torsion points and of the 4-torsion points on the given elliptic curve. %In addition, we show that such a description involves the Weil pairing in a certain way. As an application, we generalize a result of Coates on the 12-th root of the discriminant of an elliptic curve.
A simple proof is given of a sufficient condition that the square root of s is simply normal to base 2, for s not a perfect square. This relates to previous work of the author.
We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). Besides this concrete interpretation, we also prove an analogous result on configuration spaces in the Grothendieck ring of varieties, suggesting new homological stabilization conjectures for configuration spaces of the plane.
A Recursive Construction of Permutation Polynomials over $\mathbb{F}_{q^2}$ with Odd Characteristic from Rédei Functions
本文利用有理Rédei函数递归构造了奇特征$\mathbb{F}_{q^2}$上的两类置换多项式，完整刻画了其合成逆，并给出了易于验证的置换条件及若干二项式、三项式实例，发现此类多项式数量庞大。
Root Clusters and Multiclusters over Imperfect Hilbertian Fields
将根簇理论从完美域推广到一般域，引入根簇大小、多簇大小等概念，并在希尔伯特域上建立了这些广义概念的逆问题结果。
Given an odd prime $p$, we provide formulas for the Hensel lifts of polynomial roots modulo $p$, and give an explicit factorization over the ring of formal power series with integer coefficients for certain reducible polynomials whose constant term is of the form $p^w$ with $w>1$. All of our formulas are given in terms of partial Bell polynomials and rely on the inversion formula of Lagrange.
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