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21
On the factorization of polynomials with small Euclidean norm
"... this paper, we refer to the noncyclotomic part of a polynomial ..."
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this paper, we refer to the noncyclotomic part of a polynomial
On Some Polynomials Allegedly Related To The ABC Conjecture
 Acta Arith
, 1997
"... this paper is to bring your attention to the following family of polynomials. ..."
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this paper is to bring your attention to the following family of polynomials.
Diophantine problems in many variables: the rôle of additive number theory
 in Topics in Number Theory
, 1999
"... Abstract. We provide an account of the current state of knowledge concerning diophantine problems in many variables, paying attention in particular to the fundamental role played by additive number theory in establishing a large part of this body of knowledge. We describe recent explicit versions o ..."
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Abstract. We provide an account of the current state of knowledge concerning diophantine problems in many variables, paying attention in particular to the fundamental role played by additive number theory in establishing a large part of this body of knowledge. We describe recent explicit versions of the theorems of Brauer and Birch concerning the solubility of systems of forms in many variables, and establish an explicit version of Birch’s Theorem in algebraic extensions of Q. Finally, we consider the implications of recent progress on explicit versions of Brauer’s Theorem for problems concerning the solubility of systems of forms in solvable extensions, such as Hilbert’s resolvant problem. 1.
On the dichotomy of Perron numbers and betaconjugates
, 2008
"... Let β > 1 be an algebraic number. A general definition of a betaconjugate of β is proposed with respect to the analytical function fβ(z) = −1 + ∑ i i≥1 tiz associated with the Rényi βexpansion dβ(1) = 0.t1t2... of unity. From Szegö’s Theorem, we study the dichotomy problem for fβ(z), in parti ..."
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Let β > 1 be an algebraic number. A general definition of a betaconjugate of β is proposed with respect to the analytical function fβ(z) = −1 + ∑ i i≥1 tiz associated with the Rényi βexpansion dβ(1) = 0.t1t2... of unity. From Szegö’s Theorem, we study the dichotomy problem for fβ(z), in particular for β a Perron number: whether it is a rational fraction or admits the unit circle as natural boundary. The first case of dichotomy meets Boyd’s works. We introduce the study of the geometry of the betaconjugates with respect to that of the Galois conjugates by means of the ErdősTurán approach and take examples of Pisot, Salem and Perron numbers which are Parry numbers to illustrate it. We discuss the possible existence of an infinite number of betaconjugates and conjecture that all real algebraic numbers> 1, in particular Perron numbers, are in C1 ∪ C2 ∪ C3 after the classification of Blanchard/BertrandMathis.
Factorization of Trinomials over Galois Fields of Characteristic 2
, 1997
"... We study the parity of the number of irreducible factors of trinomials over Galois fields of characteristic 2. As a consequence, some sufficient conditions for a trinomial being reducible are obtained. For example, x n + ax k + b 2 GF (2 t )[x] is reducible if both n, t are even, except possibly whe ..."
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We study the parity of the number of irreducible factors of trinomials over Galois fields of characteristic 2. As a consequence, some sufficient conditions for a trinomial being reducible are obtained. For example, x n + ax k + b 2 GF (2 t )[x] is reducible if both n, t are even, except possibly when n = 2k, k odd. The case t = 1 was treated by R.G. Swan [10], who showed that x n + x k + 1 is reducible over GF (2) if 8n.
On homeomorphic product measures on the Cantor set
 AMER. MATH. SOC
, 2006
"... Let µ(r) be the Bernoulli measure on the Cantor space given as the infinite product of twopoint measures with weights r and 1 − r. It is a longstanding open problem to characterize those r and s such that µ(r) and µ(s) are topologically equivalent (i.e., there is a homeomorphism from the Cantor ..."
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Let µ(r) be the Bernoulli measure on the Cantor space given as the infinite product of twopoint measures with weights r and 1 − r. It is a longstanding open problem to characterize those r and s such that µ(r) and µ(s) are topologically equivalent (i.e., there is a homeomorphism from the Cantor space to itself sending µ(r) to µ(s)). The (possibly) weaker property of µ(r) and µ(s) being continuously reducible to each other is equivalent to a property of r and s called binomial equivalence. In this paper we define an algebraic property called “refinability ” and show that, if r and s are refinable and binomially equivalent, then µ(r) and µ(s) are topologically equivalent. We then give a class of examples of refinable numbers; in particular, the positive numbers r and s such that s = r 2 and r = 1 −s 2 are refinable, so the corresponding measures are topologically equivalent.
On a Theorem of Jordan
 BULLETIN (NEW SERIES) OF THE AMERICAN MATHEMATICAL SOCIETY
, 2003
"... The theorem of Jordan which I want to discuss here dates from 1872. It is an elementary result on finite groups of permutations. I shall first present its translations in Number Theory and Topology. ..."
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The theorem of Jordan which I want to discuss here dates from 1872. It is an elementary result on finite groups of permutations. I shall first present its translations in Number Theory and Topology.
EQUALITIES AMONG CAPACITIES OF (d, k)CONSTRAINED SYSTEMS ∗
"... Abstract. In this paper, we consider the problem ofdetermining when the capacities ofdistinct (d, k)constrained systems can be equal. A (d, k)constrained system consists ofbinary sequences which have at least d zeros and at most k zeros between any two successive ones. Ifwe let C(d, k) denote the ..."
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Abstract. In this paper, we consider the problem ofdetermining when the capacities ofdistinct (d, k)constrained systems can be equal. A (d, k)constrained system consists ofbinary sequences which have at least d zeros and at most k zeros between any two successive ones. Ifwe let C(d, k) denote the capacity ofa (d, k)constrained system, then it is known that C(d, 2d) =C(d +1, 3d +1) and C(d, 2d +1)=C(d +1, ∞). Repeated application ofthese two identities also yields the chain ofequalities C(1, 2) = C(2, 4) = C(3, 7) = C(4, ∞). We show that these are the only equalities possible among the capacities of(d, k)constrained systems. In the process, we also provide useful factorizations of the characteristic polynomials for these constraints. Key words. Shannon capacity, constrained systems, (d, k)constraints, polynomial factorization
An extension of a theorem of Ljunggren
"... this paper are essentially the same as those of Ljunggren. He presented some key ideas introducing reciprocal polynomials into the problem of determining how polynomials with small Euclidean norm factor. The proof he gave of his theorem above involved consideration of several cases depending on the ..."
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this paper are essentially the same as those of Ljunggren. He presented some key ideas introducing reciprocal polynomials into the problem of determining how polynomials with small Euclidean norm factor. The proof he gave of his theorem above involved consideration of several cases depending on the relative sizes of the exponents n, m, and p. In the case of Theorem 1 (or Theorem 2), we were able to bypass considering as many cases, mainly because the coefficients are more restrictive. We make no pretense here, however, of developing new approaches; this paper is merely a note that a five term version of Ljunggren's theorem does in fact exist. We give a proof of Theorem 1 below; a proof of Theorem 2 can be made with very few changes.
THE MINIMAL EUCLIDEAN NORM OF AN ALGEBRAIC NUMBER IS EFFECTIVELY COMPUTABLE
, 1992
"... For P 2 Z[x], let kPk denote the Euclidean norm of the coefficient vector of P . For an algebraic number ff, with minimal polynomial A, define the Euclidean norm of ff by kffk = kkAk ; where k is the smallest positive integer for which kA 2 Z[x]. Define the minimal Euclidean norm of ff by kffk ..."
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For P 2 Z[x], let kPk denote the Euclidean norm of the coefficient vector of P . For an algebraic number ff, with minimal polynomial A, define the Euclidean norm of ff by kffk = kkAk ; where k is the smallest positive integer for which kA 2 Z[x]. Define the minimal Euclidean norm of ff by kffk min = min \Phi kPk : P 2 Z[x]; P (ff) = 0; P 6j 0 \Psi : Given an algebraic number ff, we show there exists a P 2 Z[x] with P (ff) = 0 and kPk = kffk min such that the degree of P is bounded above by an explicit function of deg ff, kffk, and kffk min . As a result, we are able to prove that both P and kffk min can be effectively computed using a suitable search procedure. As an indication of the difficulties involved, we show that the determination of P is equivalent to finding a shortest nonzero vector in an infinite union of certain lattices. After introducing several techniques for reducing the search space, a practical algorithm is presented which has been successful in computi...