Root separation for polynomials with reducible derivative

Suppose f is a degree d polynomial with integer coefficients whose derivative f′ is a polynomial reducible over ℚ. We give a lower bound for the distance between two distinct roots of f in terms of d, the height H(f) of f, and the degree m of the irreducible factor of f′ with largest degree. The exponent (d + m − 1)/2 that appears as the power of H(f) is smaller than the corresponding exponent d − 1 in Mahler’s bound.

On Motivic Realizations of the Canonical Hermitian Variations of Hodge Structure of Calabi–Yau Type over type Domains

Let${\mathcal{D}}$be the irreducible Hermitian symmetric domain of type$D_{2n}^{\mathbb{H}}$. There exists a canonical Hermitian variation of real Hodge structure${\mathcal{V}}_{\mathbb{R}}$of Calabi–Yau type over ${\mathcal{D}}$. This short note concerns the problem of giving motivic realizations for ${\mathcal{V}}_{\mathbb{R}}$. Namely, we specify a descent of${\mathcal{V}}_{\mathbb{R}}$from$\mathbb{R}$to$\mathbb{Q}$and ask whether the$\mathbb{Q}$-descent of${\mathcal{V}}_{\mathbb{R}}$can be realized as sub-variation of rational Hodge structure of those coming from families of algebraic varieties. When$n=2$, we give a motivic realization for ${\mathcal{V}}_{\mathbb{R}}$. When$n\geqslant 3$, we show that the unique irreducible factor of Calabi–Yau type in$\text{Sym}^{2}{\mathcal{V}}_{\mathbb{R}}$can be realized motivically.

Structure of transition classes for factor codes on shifts of finite type

Given a factor code ${\it\pi}$ from a shift of finite type $X$ onto a sofic shift $Y$, the class degree of ${\it\pi}$ is defined to be the minimal number of transition classes over the points of $Y$. In this paper, we investigate the structure of transition classes and present several dynamical properties analogous to the properties of fibers of finite-to-one factor codes. As a corollary, we show that for an irreducible factor triple, there cannot be a transition between two distinct transition classes over a right transitive point, answering a question raised by Quas.

A Gelfand Model for Weyl Groups of Type D2n

A Gelfand model for a finite group G is a complex representation of G, which is isomorphic to the direct sum of all irreducible representations of G. When G is isomorphic to a subgroup of GLn(ℂ), where ℂ is the field of complex numbers, it has been proved that each G-module over ℂ is isomorphic to a G-submodule in the polynomial ring ℂ[x1,…,xn], and taking the space of zeros of certain G-invariant operators in the Weyl algebra, a finite-dimensional G-space 𝒩G in ℂ[x1,…,xn] can be obtained, which contains all the simple G-modules over ℂ. This type of representation has been named polynomial model. It has been proved that when G is a Coxeter group, the polynomial model is a Gelfand model for G if, and only if, G has not an irreducible factor of type D2n, E7, or E8. This paper presents a model of Gelfand for a Weyl group of type D2n whose construction is based on the same principles as the polynomial model.

Word equations in a uniquely divisible group

International audience We study equations in groups $G$ with unique $m$-th roots for each positive integer $m$. A word equation in two letters is an expression of the form$ w(X,A) = B$, where $w$ is a finite word in the alphabet ${X,A}$. We think of $A,B ∈G$ as fixed coefficients, and $X ∈G$ as the unknown. Certain word equations, such as $XAXAX=B$, have solutions in terms of radicals: $X = A^-1/2(A^1/2BA^1/2)^1/3A^-1/2$, while others such as $X^2 A X = B$ do not. We obtain the first known infinite families of word equations not solvable by radicals, and conjecture a complete classification. To a word w we associate a polynomial $P_w ∈ℤ[x,y]$ in two commuting variables, which factors whenever $w$ is a composition of smaller words. We prove that if $P_w(x^2,y^2)$ has an absolutely irreducible factor in $ℤ[x,y]$, then the equation $w(X,A)=B$ is not solvable in terms of radicals. Nous étudions des équations dans les groupes $G$ avec les $m$-th racines uniques pour chaque nombre entier positif m. Une équation de mot dans deux lettres est une expression de la forme $w(X, A) = B$, où $w$ est un mot fini dans l'alphabet ${X, A}$. Nous pensons $A, B ∈G$ en tant que coefficients fixes, et $X ∈G$ en tant que inconnu. Certaines équations de mot, telles que $XAXAX=B$, ont des solutions en termes de radicaux: $X = A^-1/2(A^1/2BA^1/2)^1/3A^-1/2$, alors que d'autres tel que $X^2 A X = B$ ne font pas. Nous obtenons les familles infinies d'abord connues des équations de mot non solubles par des radicaux, et conjecturons une classification complété. Á un mot $w$ nous associons un polynôme $P_w ∈ℤ[x, y]$ dans deux variables de permutation, qui factorise toutes les fois que $w$ est une composition de plus petits mots. Nous montrons que si $P_w(x^2, y^2)$ a un facteur absolument irréductible dans $ℤ[x, y]$, alors l'équation $w(X, A)=B$ n'est pas soluble en termes de radicaux.

On the distribution of reducible polynomials

Let ℛn(t) denote the set of all reducible polynomials p(X) over ℤ with degree n ≥ 2 and height ≤ t. We determine the true order of magnitude of the cardinality |ℛn(t)| of the set ℛn(t) by showing that, as t → ∞, t 2 log t ≪ |ℛ2(t)| ≪ t 2 log t and t n ≪ |ℛn(t)| ≪ t n for every fixed n ≥ 3. Further, for 1 < n/2 < k < n fixed let ℛk,n(t) ⊂ ℛn(t) such that p(X) ∈ ℛk,n(t) if and only if p(X) has an irreducible factor in ℤ[X] of degree k. Then, as t → ∞, we always have t k+1 ≪ |ℛk,n(t)| ≪ t k+1 and hence |ℛn−1,n (t)| ≫ |ℛn(t)| so that ℛn−1,n (t) is the dominating subclass of ℛn(t) since we can show that |ℛn(t)∖ℛn−1,n (t)| ≪ t n−1(log t)2.On the contrary, if R ns(t) is the total number of all polynomials in ℛn(t) which split completely into linear factors over ℤ, then t 2(log t)n−1 ≪ R ns(t) ≪ t 2 (log t)n−1 (t → ∞) for every fixed n ≥ 2.

Every Real Algebraic Integer Is a Difference of Two Mahler Measures

We prove that every real algebraic integer α is expressible by a difference of two Mahler measures of integer polynomials. Moreover, these polynomials can be chosen in such a way that they both have the same degree as that of α, say d, one of these two polynomials is irreducible and another has an irreducible factor of degree d, so that α = M(P)−bM(Q) with irreducible polynomials P,Q ∈ ℤ[X] of degree d and a positive integer b. Finally, if d ⩽ 3, then one can take b = 1.

Maurice Godelier and the Metamorphosis of Kinship, A Review Essay

Once it was thought that kinship was the preeminent subject of anthropology, one about which considerable progress was possible. “Kinship” itself was, for some, fairly unproblematic. Thus Radcliffe-Brown (1952: 46) asserted that, “if any society establishes a system of corporations on the basis of kinship … it must necessarily adopt a system of unilineal reckoning of succession,” and Fortes (1959: 209) announced that, “Kinship, being an irreducible factor in social structure has an axiomatic validity.” However, in the late 1960s and early 1970s one leading figure of anthropology after the other declared that there really was no such thing as kinship. “The process of making kinship into a single theoretical entity seems to me no better than the invention of ‘totemism’” (Terray 1969: 135–36; 1972: 140–41). “There is no such thing as kinship, and it follows that there can be no such thing as kinship theory” (Needham 1971: 5). “‘Kinship,’ like totemism, the matrilineal complex and matriarchy, is a non-subject since it does not exist in any culture known to man” (Schneider 1972: 59). “The whole notion of ‘a kinship system’ as an isolable structure of sentiments, norms, or categorical distinctions is misleading because it assumes, or seems to assume, that the ordering principles of a society are partitionable into natural kinds only adventitiously connected” (Geertz and Geertz 1975: 156). For various political and intellectual reasons, “kinship” appeared to many to have died out as an area of analytic interest within anthropology during the 1970s and 1980s, despite many indications to the contrary. Now Godelier has made a major effort to revive attention to matters usually bunched under the phrase “kinship,” and, at least as concerns French popular taste, seems largely to have succeeded. For him kinship has not died, but instead transformed itself both in fact and “theoretically.”