Let p > 2 be a prime, let n > m > 0. Let pi_n be the norm of zeta_{p^n} - 1 under C_{p-1}, so that Z_(p)[pi_n] | Z_(p) is a purely ramified extension of discrete valuation rings of degree p^{n-1}. The minimal polynomial of pi_n over Q(pi_m) is an Eisenstein polynomial; we give lower bounds for its coefficient valuations at pi_m. The function field analogue, as introduced by Carlitz and Hayes, is studied as well.
We prove R = T theorems for certain reducible residual Galois representations. We answer in the positive a question of Gross and Lubin on whether certain Hecke algebras T are discrete valuation rings. In order to prove these results we determine (using the theory of Breuil modules) when two finite flat group schemes G and H of order p over an arbitrarily tamely ramified discrete valuation ring admit an extension not killed by p.
We study polynomials with integer coefficients which become Eisenstein polynomials after the additive shift of a variable. We call such polynomials shifted Eisenstein polynomials. We determine an upper bound on the maximum shift that is needed given a shifted Eisenstein polynomial and also provide a lower bound on the density of shifted Eisenstein polynomials, which is strictly greater than the density of classical Eisenstein polynomials. We also show that the number of irreducible degree $n$ polynomials that are not shifted Eisenstein polynomials is infinite. We conclude with some numerical r…
In this paper we compute the density of monic and non-monic Eisenstein polynomials of fixed degree having entries in an integrally closed subring of a function field over a finite field.
In this note, we give a criteria whether given two Eisenstein polynomials over a padic field define the same extension (Proposition 1.6). In particular, we completely identify Eisenstein polynomials of degree p (Theorem 1.16). This note is an English translation of a part of the original paper [12].
We give a criterion whether given Eisenstein polynomials over a local field K define the same extension over K in terms of a certain non-Archimedean metric on the set of polynomials. The criterion and its proof depend on ramification theory.
Previously Heyman and Shparlinski gave an asymptotic formula with error term for the number of Eisenstein polynomials of fixed degree and bounded height. Let $ψ(f)$ denote the number of primes for which a polynomial $f$ is Eisenstein. We give expressions for the mean and variance of the function $ψ$ for each fixed degree, where the polynomials are ordered according to their height.
We obtain a more precise version of an asymptotic formula of A. Dubickas for the number of monic Eisenstein polynomials of fixed degree $d$ and of height at most $H$, as $H\to \infty$. In particular, we give an explicit bound for the error term. We also obtain an asymptotic formula for arbitrary Eisenstein polynomials of height at most $H$.
In this paper we provide a complete answer to a question by Heyman and Shparlinski concerning the natural density of polynomials which are irreducible by Eisenstein's criterion after applying some shift. The main tool we use is a local to global principle for density computations over a free $\mathbb Z$-module of finite rank.
Let $E_k$ be the normalized Eisenstein series of weight $k$ on $\mathrm{SL}_{2}(\mathbb{Z})$. Let $φ_k$ be the polynomial that encodes the $j$-invariants of non-elliptic zeros of $E_k$. In 2001, Gekeler observed that the polynomials $φ_k$ seem to be irreducible (and verified this claim for $k\leq 446$). We show that $φ_k$ is irreducible for infinitely many $k$.
In this paper, based on Serre's $p$-adic family of Eisenstein series, we prove a general family of congruences for Eisenstein series $G_k$ in the form $$ \sum_{i=1}^n g_i(p)G_{f_i(p)}\equiv g_0(p)\mod p^N, $$ where $f_1(t),\ldots,f_n(t)\in\mathbb{Z}[t]$ are non-constant integer polynomials with positive leading coefficients and $g_0(t),\ldots,g_n(t)\in\mathbb{Q}(t)$ are rational functions. This generalizes the classical von Staudt-Clausen's and Kummer's congruences of Eisenstein series, and also yields some new congruences.
Let $K$ be a valued field (in general $K$ is not heselian) with valuation $v$ and $A(x)\in K[x]$ be a polynomial of degree $n$. We find necessary and sufficient conditions for the existence of the elements $s,t,u\in K$, $s\neq 0\neq u$, such that at least one of the polynomials $u^nA(\frac{sx+t}{u})$, $(tx+u)^nA(\frac{sx}{tx+u})$, $(ux)^nA(\frac{tx+s}{ux})$ or $(ux+t)^nA(\frac{s}{ux+t})$ is an Eisenstein-Dumas polynomial at $v$, provided that the characteristic of the residue field of $v$ does not divide $n$. Furthermore, we show that if the orbit $A(x)GL(2,K)$ contains an Eisenstein-Dumas pol…
Eisenstein polynomials, which were defined by Oura, are analogues of the concept of an Eisenstein series. Oura conjectured that there exist some analogous properties between Eisenstein series and Eisenstein polynomials. In this paper, we provide new analogous properties of Eisenstein polynomials and zeta polynomials. These properties are finite analogies of certain properties of Eisenstein series.
Let $K$ be a local field with finite residue field, we define a normal form for Eisenstein polynomials depending on the choice of a uniformizer $π_K$ and of residue representatives. The isomorphism classes of extensions generated by the polynomials in the family exhaust all totally ramified extensions, and the multiplicity with which each isomorphism class $L/K$ appears is always smaller than the number of conjugates of $L$ over $K$. An algorithm to recover the set of all special polynomials generating the extension determined by a general Eisenstein polynomial is described. We also give a cri…
Let $p\neq2$ be a prime. We show a technique based on local class field theory and on the expansions of certain resultants which allows to recover very easily Lbekkouri's characterization of Eisenstein polynomials generating cyclic wild extensions of degree $p^2$ over $\Q_p$, and to extend it to the case of the base field $K$ being an unramified extension of $\Q_p$. Furthermore, when a polynomial satisfies only some of the stated conditions, we show that the first unsatisfied condition gives information about the Galois group of the normal closure. This permits to give a complete classificatio…
We solve Lehmer's problem for a class of polynomials arising from Hermitian matrices over the Eisenstein and Gaussian integers, that is, we show that all such polynomials have Mahler measure at least Lehmer's number τ_0 = 1.17628... .
Let $K$ be a local field with residue characteristic $p$ and let $L/K$ be a totally ramified extension of degree $p^k$. In this paper we show that if $L/K$ has only two distinct indices of inseparability then there exists a uniformizer $π_L$ for $L$ whose minimum polynomial over $K$ has at most three terms. This leads to an explicit classification of extensions with two indices of inseparability. Our classification extends work of Amano, who considered the case $k=1$.
Eisenstein polynomials, which were defined by the second author, are analogues of the concept of an Eisenstein series. The second author conjectured that there exist some analogous properties between Eisenstein series and Eisenstein polynomials. In the previous paper, the first author provided new analogous properties of Eisenstein polynomials and zeta polynomials for the Type II case. In this paper, the analogous properties of Eisenstein polynomials and zeta polynomials are shown to also hold for the Type I, Type III, and Type IV cases. These properties are finite analogies of certain propert…
Let $K$ be a local field whose residue field has characteristic $p$ and let $L/K$ be a finite separable totally ramified extension. Let $π_L$ be a uniformizer for $L$ and let $f(X)$ be the minimum polynomial for $π_L$ over $K$. Suppose $\tildeπ_L$ is another uniformizer for $L$ such that $\tildeπ_L\equivπ_L+rπ_L^{\ell+1} \pmod{π_L^{\ell+2}}$ for some $\ell\ge1$ and $r\in O_K$. Let $\tilde{f}(X)$ be the minimum polynomial for $\tildeπ_L$ over $K$. In this paper we give congruences for the coefficients of $\tilde{f}(X)$ in terms of $r$ and the coefficients of $f(X)$. These congruences improve an…
The famous irreducibility criteria of Schönemann-Eisenstein and Dumas rely on information on the divisibility of the coefficients of a polynomial by a single prime number. In this paper we provide several irreducibility criteria of Schönemann-Eisenstein-Dumas-type for polynomials with integer coefficients, criteria that are given by some divisibility conditions for their coefficients with respect to arbitrarily many prime numbers. A special attention will be paid to those irreducibility criteria that require information on the divisibility of the coefficients by two distinct prime numbers.
Under certain conditions, we prove that the set of zeros of an elliptic net forms an Abelian group. We present two applications of this fact. Firstly we give a generalization of a theorem of Ayad on valuations of division polynomials in the context of net polynomials. Secondly we generalize a theorem of Ward on symmetry of elliptic divisibility sequences to the case of elliptic nets.
We describe a decomposition algorithm for elliptic multiple zeta values, which amounts to the construction of an injective map $ψ$ from the algebra of elliptic multiple zeta values to a space of iterated Eisenstein integrals. We give many examples of this decomposition, and conclude with a short discussion about the image of $ψ$. It turns out that the failure of surjectivity of $ψ$ is in some sense governed by period polynomials of modular forms.
For a prime $p$ and an integer $x$, the $p$-adic valuation of $x$ is denoted by $ν_{p}(x)$. For a polynomial $Q$ with integer coefficients, the sequence of valuations $ν_{p}(Q(n))$ is shown to be either periodic or unbounded. The first case corresponds to the situation where $Q$ has no roots in the ring of $p$-adic integers. In the periodic situation, the period length is determined.
Given integers $\ell > m >0$, we define monic polynomials $X_n$, $Y_n$, and $Z_n$ with the property that $μ$ is a zero of $X_n$ if and only if the triple $(μ,μ+m,μ+\ell)$ satisfies $x^n + y^n = z^n$. It is shown that the irreducibility of these polynomials implies Fermat's last theorem. It is also shown, in a precise asymptotic sense, that for a vast majority of cases, these polynomials are irreducible via Eisenstein's criterion. We conclude by offering a conjecture on powerful numbers.
Let $π$ traverse a sequence of cuspidal automorphic representations of GL(2) with large prime level, unramified central character and bounded infinity type. For G either of the groups GL(1) or PGL(2), let H(G) denote the assertion that subconvexity holds for G-twists of the adjoint $L$-function of $π$, with polynomial dependence upon the conductor of the twist. We show that H(GL(1)) implies H(PGL(2)). In geometric terms, H(PGL(2)) corresponds roughly to an instance of arithmetic quantum unique ergodicity with a power savings in the error term, H(GL(1)) to the special case in which the relevant…
In this paper we compute the 2-adic valuations of some polynomials associated with the definite integral $\int_{0}^{\infty} \frac{dx}{(x^4+2*a*x^2+1)^(m+1)}$
Let $K$ be a number field defined by a monic irreducible polynomial $F(X) \in \mathbb{Z}[X]$, $p$ a fixed rational prime, and $ν_p$ the discrete valuation associated to $p$. Assume that $\overline{F}(X)$ factors modulo $p$ into the product of powers of $r$ distinct monic irreducible polynomials. We present in this paper a condition, weaker than the known ones, which guarantees the existence of exactly $r$ valuations of $K$ extending $ν_p$. We further specify the ramification indices and residue degrees of these extended valuations in such a way that generalizes the known estimates. Some useful…
Assuming Lang's conjectured lower bound on the heights of non-torsion points on an elliptic curve, we show that there exists an absolute constant C such that for any elliptic curve E/Q and non-torsion point P in E(Q), there is at most one integral multiple [n]P such that n > C. The proof is a modification of a proof of Ingram giving an unconditional but not uniform bound. The new ingredient is a collection of explicit formulae for the sequence of valuations of the division polynomials. For P of non-singular reduction, such sequences are already well described in most cases, but for P of singul…
We consider, for even $s$, the secant Dirichlet series $ψ_s (τ) = \sum_{n = 1}^{\infty} \frac{\sec (πn τ)}{n^s}$, recently introduced and studied by Lal\'ın, Rodrigue and Rogers. In particular, we show, as conjectured and partially proven by Lal\'ın, Rodrigue and Rogers, that the values $ψ_{2 m} ( \sqrt{r})$, with $r > 0$ rational, are rational multiples of $π^{2 m}$. We then put the properties of the secant Dirichlet series into context by showing that they are Eichler integrals of odd weight Eisenstein series of level $4$. This leads us to consider Eichler integrals of general Eisenstein ser…
The following result, a consequence of Dumas criterion for irreducibility of polynomials over integers, is generally proved using the notion of Newton diagram: Let $f(x)$ be a polynomial with integer coefficients and $k$ be a positive integer relatively prime to the degree of $f(x)$. Suppose that there exists a prime number $p$ such that the leading coefficient of $f(x)$ is not divisible by $p$, all the remaining coefficients are divisible by $p^k$, and the constant term of $f(x)$ is not divisible by $p^{k+1}$. Then $f(x)$ is irreducible over $\mathbb{Z}$. For $k=1$, this is precisely the Eise…
Let $(K,v)$ be a henselian valued field. Let $\mathbb{P}^{dless}\subset K[x]$ be the set of monic, irreducible polynomials which are defectless and have degree greater than one. For a certain equivalence relation $\,\approx\,$ on $\,\mathbb{P}^{dless}$, we establish a canonical bijection $\mathbb{M}\to \mathbb{P}^{dless}/\!\!\approx$, where $\mathbb{M}$ is a discrete MacLane space, constructed in terms of inductive valuations on $K[x]$ extending $v$.
Let t[n] be a sequence that satisfies a first order homogeneous recurrence t[n] = Q[n]*t[n-1], where Q is a polynomial with integer coefficients. The asymptotic behavior of the p-adic valuation of t[n] is described under the assumption that all the roots of Q in Z/pZ have nonvanishing derivative.
Let $(K,v)$ be a valued field, and $μ$ an inductive valuation on $K[x]$ extending $v$. Let $G_μ$ be the graded algebra of $μ$ over $K[x]$, and $κ$ the maximal subfield of the subring of $G_μ$ formed by the homogeneous elements of degree zero. In this paper, we find an algorithm to compute the field $κ$ and the residual polynomial operator $R_μ: K[x]\toκ[y]$, where $y$ is another indeterminate, without any need to perform computations in the graded algebra. This leads to an OM algorithm to compute the factorization of separable defectless polynomials over henselian fields.
L. Szpiro and T. Tucker recently proved that under mild conditions, the valuation of the minimal discriminant of an elliptic curve with semistable reduction over a discrete valuation ring can be expressed in terms of intersections between n-torsion and 2-torsion, where n tends to infinity. The argument of Szpiro and Tucker is geometric in nature. We give a proof based on the arithmetic of division polynomials, and generalize the result to the case of hyperelliptic curves.
Properties of 2-adic valuation sequences for general quadratic polynomials with integer coefficients are determined directly from the coefficients. These properties include boundedness or unboundedness, periodicity, and valuations at terminating nodes. We completely describe the periodic sequences in the bounded case. Throughout, we frame results in terms of trees and sequences.
For a discrete valuation domain $V$ with maximal ideal $\mathfrak{m}$ such that the residue field $V/\mathfrak{m}$ is finite, there exists a sequence of polynomials $(F_n(x))_{n \ge 0}$ defined over the quotient field $K$ of $V$ that forms a basis of the $V$-module $\text{Int}(V) = \{f \in K[x] | f(V)\subseteq V\}$. This sequence of polynomials bears many resemblances to the classical binomial polynomials $(\binom{x}{n})_{n \ge 0}$. In this paper, we introduce a generating polynomial to account for the distribution of the $V$-values of the polynomials $F_n(x)$ modulo the maximal ideal $\mathfr…
Let $k$ be a locally compact complete field with respect to a discrete valuation $v$. Let $\oo$ be the valuation ring, $\m$ the maximal ideal and $F(x)\in\oo[x]$ a monic separable polynomial of degree $n$. Let $δ=v(\dsc(F))$. The Montes algorithm computes an OM factorization of $F$. The single-factor lifting algorithm derives from this data a factorization of $F \md{\m^ν}$, for a prescribed precision $ν$. In this paper we find a new estimate for the complexity of the Montes algorithm, leading to an estimation of $O(n^{2+ε}+n^{1+ε}δ^{2+ε}+n^2ν^{1+ε})$ word operations for the complexity of the c…
In this paper, we study factorizations in the additive monoids of positive algebraic valuations $\mathbb{N}_0[α]$ of the semiring of polynomials $\mathbb{N}_0[X]$ using a methodology introduced by D. D. Anderson, D. F. Anderson, and M. Zafrullah in 1990. A cancellative commutative monoid is atomic if every non-invertible element factors into irreducibles. We begin by determining when $\mathbb{N}_0[α]$ is atomic, and we give an explicit description of its set of irreducibles. An atomic monoid is a finite factorization monoid (FFM) if every element has only finitely many factorizations (up to or…
The Shub-Smale Tau Conjecture is a hitherto unproven statement (on integer roots of polynomials) whose truth implies both a variant of $P\neq NP$ (for the BSS model over C) and the hardness of the permanent. We give alternative conjectures, some potentially easier to prove, whose truth still implies the hardness of the permanent. Along the way, we discuss new upper bounds on the number of $p$-adic valuations of roots of certain sparse polynomial systems, culminating in a connection between quantitative p-adic geometry and complexity theory.
Lie algebras for multiple Eisenstein series and multiple q-zeta values
Graduate School of Mathematics, Nagoya University（名古屋大学数学研究科）；Department of Mathematics, Graduate School of Science, Tokyo University of Science（东京理科大学理学部数学系）
We address the problem of the stability of the computations of resultants and subresultants of polynomials defined over complete discrete valuation rings (e.g. Zp or k[[t]] where k is a field). We prove that Euclide-like algorithms are highly unstable on average and we explain, in many cases, how one can stabilize them without sacrifying the complexity. On the way, we completely determine the distribution of the valuation of the principal subresultants of two random monic p-adic polynomials having the same degree.
Let $p$ denote an odd prime. In this paper, we are concerned with the $p$-divisibility of additive exponential sums associated to one variable polynomials over a finite field of characteristic $p$, and with (the very close question of) determining the Newton polygons of some families of Artin-Schreier curves, i.e. $p$-cyclic coverings of the projective line in characteristic $p$. We first give a lower bound on the $p$-divisibility of exponential sums associated to polynomials of fixed degree. Then we show that an Artin-Schreier curve defined over a finite field of characteristic $p$ cannot be …
