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We prove that for every sufficiently large integer
$n$
, the polynomial
$1+x+x^{2}/11+x^{3}/111+\cdots +x^{n}/111\ldots 1$
is irreducible over the rationals, where the coefficient of
$x^{k}$
for
$1\leqslant k\leqslant n$
is the reciprocal of the decimal number consisting of
$k$
digits which are each
$1$
. Similar results following from the same techniques are discussed.
A polynomial
$f$
over a finite field
$\mathbb{F}_{q}$
can be classified as a permutation polynomial by the Hermite–Dickson criterion, which consists of conditions on the powers
$f^{e}$
for each
$e$
from
$1$
to
$q-2$
, as well as the existence of a unique solution to
$f(x)=0$
in
$\mathbb{F}_{q}$
. Carlitz and Lutz gave a variant of the criterion. In this paper, we provide an alternate proof to the theorem of Carlitz and Lutz.
The probability of successfully spending twice the same bitcoins is considered. A double-spending attack consists in issuing two transactions transferring the same bitcoins. The first transaction, from the fraudster to a merchant, is included in a block of the public chain. The second transaction, from the fraudster to himself, is recorded in a block that integrates a private chain, exact copy of the public chain up to substituting the fraudster-to-merchant transaction by the fraudster-to-fraudster transaction. The double-spending hack is completed once the private chain reaches the length of the public chain, in which case it replaces it. The growth of both chains are modelled by two independent counting processes. The probability distribution of the time at which the malicious chain catches up with the honest chain, or, equivalently, the time at which the two counting processes meet each other, is studied. The merchant is supposed to await the discovery of a given number of blocks after the one containing the transaction before delivering the goods. This grants a head start to the honest chain in the race against the dishonest chain.
A cyclotomic polynomial
$\unicode[STIX]{x1D6F7}_{k}(x)$
is an essential cyclotomic factor of
$f(x)\in \mathbb{Z}[x]$
if
$\unicode[STIX]{x1D6F7}_{k}(x)\mid f(x)$
and every prime divisor of
$k$
is less than or equal to the number of terms of
$f.$
We show that if a monic polynomial with coefficients from
$\{-1,0,1\}$
has a cyclotomic factor, then it has an essential cyclotomic factor. We use this result to prove a conjecture posed by Mercer [‘Newman polynomials, reducibility, and roots on the unit circle’, Integers12(4) (2012), 503–519].
Given
$f\in \mathbb{Z}[t]$
of positive degree, we investigate the existence of auxiliary polynomials
$g\in \mathbb{Z}[t]$
for which
$f(g(t))$
factors as a product of polynomials of small relative degree. One consequence of this work shows that for any quadratic polynomial
$f\in \mathbb{Z}[t]$
and any
$\unicode[STIX]{x1D700}>0$
, there are infinitely many
$n\in \mathbb{N}$
for which the largest prime factor of
$f(n)$
is no larger than
$n^{\unicode[STIX]{x1D700}}$
.
We generalize known results about Hilbertian fields to Hilbertian rings. For example, let R be a Hilbertian ring (e.g. R is the ring of integers of a number field) with quotient field K and let A be an abelian variety over K. Then, for every extension M of K in K(Ator(Ksep)), the integral closure RM of R in M is Hilbertian.
We revisit the coordinatisation method for projective planes by considering the consequences of using finite fields to coordinatise projective planes of prime power order. This leads to some general restrictions on the form of the resulting planar ternary ring (PTR) when viewed as a trivariate polynomial over the field. We also consider how the Lenz–Barlotti type of the plane being coordinatised impacts the form of the PTR polynomial, thereby deriving further restrictions.
Let K be a global field,
$\mathcal{V}$
a proper subset of the set of all primes of K,
$\mathcal{S}$
a finite subset of
$\mathcal{V}$
, and
${\tilde K}$
(resp. Ksep) a fixed algebraic (resp. separable algebraic) closure of K with
$K_\mathrm{sep}\{\subseteq}{\tilde K}$
. Let Gal(K) = Gal(Ksep/K) be the absolute Galois group of K. For each
$\mathfrak{p}\in\mathcal{V}$
, we choose a Henselian (respectively, a real or algebraic) closure
$K_\mathfrak{p}$
of K at
$\mathfrak{p}$
in
${\tilde K}$
if
$\mathfrak{p}$
is non-archimedean (respectively, archimedean). Then,
$K_{\mathrm{tot},\mathcal{S}}=\bigcap_{\mathfrak{p}\in\mathcal{S}}\bigcap_{\tau\in{\rm Gal}(K)}K_\mathfrak{p}^\tau$
is the maximal Galois extension of K in Ksep in which each
$\mathfrak{p}\in\mathcal{S}$
totally splits. For each
$\mathfrak{p}\in\mathcal{V}$
, we choose a
$\mathfrak{p}$
-adic absolute value
$|~|_\mathfrak{p}$
of
$K_\mathfrak{p}$
and extend it in the unique possible way to
${\tilde K}$
. Finally, we denote the compositum of all symmetric extensions of K by Ksymm. We consider an affine absolutely integral variety V in
$\mathbb{A}_K^n$
. Suppose that for each
$\mathfrak{p}\in\mathcal{S}$
there exists a simple
$K_\mathfrak{p}$
-rational point
$\mathbf{z}_\mathfrak{p}$
of V and for each
$\mathfrak{p}\in\mathcal{V}\smallsetminus\mathcal{S}$
there exists
$\mathbf{z}_\mathfrak{p}\in V({\tilde K})$
such that in both cases
$|\mathbf{z}_\mathfrak{p}|_\mathfrak{p}\le1$
if
$\mathfrak{p}$
is non-archimedean and
$|\mathbf{z}_\mathfrak{p}|_\mathfrak{p}<1$
if
$\mathfrak{p}$
is archimedean. Then, there exists
$\mathbf{z}\in V(K_{\mathrm{tot},\mathcal{S}}\cap K_\mathrm{symm})$
such that for all
$\mathfrak{p}\in\mathcal{V}$
and for all τ ∈ Gal(K), we have
$|\mathbf{z}^\tau|_\mathfrak{p}\le1$
if
$\mathfrak{p}$
is archimedean and
$|\mathbf{z}^\tau|_\mathfrak{p}<1$
if
$\mathfrak{p}$
is non-archimedean. For
$\mathcal{S}=\emptyset$
, we get as a corollary that the ring of integers of Ksymm is Hilbertian and Bezout.
In this paper we consider the integral functionals of the general epidemic model up to its extinction. We develop a new approach to determine the exact Laplace transform of such integrals. In particular, we obtain the Laplace transform of the duration of the epidemic T, the final susceptible size ST, the area under the trajectory of the infectives AT, and the area under the trajectory of the susceptibles BT. The method relies on the construction of a family of martingales and allows us to solve simple recursive relations for the involved parameters. The Laplace transforms are then expanded in terms of a special class of polynomials. The analysis is generalized in part to Markovian epidemic processes with arbitrary state-dependent rates.
The algebraic proof of the fundamental theorem of algebra uses two facts about real numbers. First, every polynomial with odd degree and real coefficients has a real root. Second, every nonnegative real number has a square root. Shipman [‘Improving the fundamental theorem of algebra’, Math. Intelligencer29(4) (2007), 9–14] showed that the assumption about odd degree polynomials is stronger than necessary; any field in which polynomials of prime degree have roots is algebraically closed. In this paper, we give a simpler proof of this result of Shipman.
Let
$K$
be a finitely generated extension of
$\mathbb{Q}$
, and let
$A$
be a nonzero abelian variety over
$K$
. Let
$\tilde{K}$
be the algebraic closure of
$K$
, and let
$\text{Gal}(K)=\text{Gal}(\tilde{K}/K)$
be the absolute Galois group of
$K$
equipped with its Haar measure. For each
$\unicode[STIX]{x1D70E}\in \text{Gal}(K)$
, let
$\tilde{K}(\unicode[STIX]{x1D70E})$
be the fixed field of
$\unicode[STIX]{x1D70E}$
in
$\tilde{K}$
. We prove that for almost all
$\unicode[STIX]{x1D70E}\in \text{Gal}(K)$
, there exist infinitely many prime numbers
$l$
such that
$A$
has a nonzero
$\tilde{K}(\unicode[STIX]{x1D70E})$
-rational point of order
$l$
. This completes the proof of a conjecture of Geyer–Jarden from 1978 in characteristic 0.
We generalize the
$\mathbb{Z}/p$
metabelian birational
$p$
-adic section conjecture for curves, as introduced and proved in Pop [On the birational
$p$
-adic section conjecture, Compos. Math. 146 (2010), 621–637], to all complete smooth varieties, provided
$p>2$
. The condition
$p>2$
seems to be of technical nature only, and might be removable.
In 2013, Weintraub gave a generalization of the classical Eisenstein irreducibility criterion in an attempt to correct a false claim made by Eisenstein. Using a different approach, we prove Weintraub's result with a weaker hypothesis in a more general setup that leads to an extension of the generalized Schönemann irreducibility criterion for polynomials with coefficients in arbitrary valued fields.
We study the question of which Henselian fields admit definable Henselian valuations (with or without parameters). We show that every field that admits a Henselian valuation with non-divisible value group admits a parameter-definable (non-trivial) Henselian valuation. In equicharacteristic 0, we give a complete characterization of Henselian fields admitting a parameter-definable (non-trivial) Henselian valuation. We also obtain partial characterization results of fields admitting -definable (non-trivial) Henselian valuations. We then draw some Galois-theoretic conclusions from our results.
We determine several variants of the classical interpolation formula for finite fields which produce polynomials that induce a desirable mapping on the nonspecified elements, and without increasing the number of terms in the formula. As a corollary, we classify those permutation polynomials over a finite field which are their own compositional inverse, extending work of C. Wells.
In this paper, we construct several new permutation polynomials over finite fields. First, using the linearised polynomials, we construct the permutation polynomial of the form
${ \mathop{\sum }\nolimits}_{i= 1}^{k} ({L}_{i} (x)+ {\gamma }_{i} ){h}_{i} (B(x))$
over
${\mathbf{F} }_{{q}^{m} } $
, where
${L}_{i} (x)$
and
$B(x)$
are linearised polynomials. This extends a theorem of Coulter, Henderson and Matthews. Consequently, we generalise a result of Marcos by constructing permutation polynomials of the forms
$xh({\lambda }_{j} (x))$
and
$xh({\mu }_{j} (x))$
, where
${\lambda }_{j} (x)$
is the
$j$
th elementary symmetric polynomial of
$x, {x}^{q} , \ldots , {x}^{{q}^{m- 1} } $
and
${\mu }_{j} (x)= {\mathrm{Tr} }_{{\mathbf{F} }_{{q}^{m} } / {\mathbf{F} }_{q} } ({x}^{j} )$
. This answers an open problem raised by Zieve in 2010. Finally, by using the linear translator, we construct the permutation polynomial of the form
${L}_{1} (x)+ {L}_{2} (\gamma )h(f(x))$
over
${\mathbf{F} }_{{q}^{m} } $
, which extends a result of Kyureghyan.
Let
${ \mathbb{F} }_{q} $
be the finite field of characteristic
$p$
containing
$q= {p}^{r} $
elements and
$f(x)= a{x}^{n} + {x}^{m} $
, a binomial with coefficients in this field. If some conditions on the greatest common divisor of
$n- m$
and
$q- 1$
are satisfied then this polynomial does not permute the elements of the field. We prove in particular that if
$f(x)= a{x}^{n} + {x}^{m} $
permutes
${ \mathbb{F} }_{p} $
, where
$n\gt m\gt 0$
and
$a\in { \mathbb{F} }_{p}^{\ast } $
, then
$p- 1\leq (d- 1)d$
, where
$d= \gcd (n- m, p- 1)$
, and that this bound of
$p$
, in terms of
$d$
only, is sharp. We show as well how to obtain in certain cases a permutation binomial over a subfield of
${ \mathbb{F} }_{q} $
from a permutation binomial over
${ \mathbb{F} }_{q} $
.
In this paper we solve the equation f(g(x))=f(x)hm(x) where f(x), g(x) and h(x) are unknown polynomials with coefficients in an arbitrary field K, f(x) is nonconstant and separable, deg g≥2, the polynomial g(x) has nonzero derivative g′(x)≠0 in K[x] and the integer m≥2 is not divisible by the characteristic of the field K. We prove that this equation has no solutions if deg f≥3 . If deg f=2 , we prove that m=2 and give all solutions explicitly in terms of Chebyshev polynomials. The Diophantine applications for such polynomials f(x) , g(x) , h(x) with coefficients in ℚ or ℤ are considered in the context of the conjecture of Cassaigne et al. on the values of Liouville’s λ function at points f(r) , r∈ℚ.
Let R be a domain contained in a rank-1 valuation ring of its quotient field. Let R⟦X⟧ be the ring of formal power series over R, and let F be the quotient field of R⟦X⟧. We prove that F is Hilbertian. This resolves and generalizes an open problem of Jarden, and allows to generalize previous Galois-theoretic results over fields of power series.
In this article we introduce and prove a ℤ/p meta-abelian form of the birationalp-adic section conjecture for curves. This is a much stronger result than the usual p-adic birational section conjecture for curves, and makes an effective p-adic section conjecture for curves quite plausible.