Polynomials with Special Regard to Reducibility

Schinzel, Andrzej

doi:10.1017/cbo9780511542916

Cited by 248 publications

(215 citation statements)

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“…Since λ k > 1 is an integer, by the theorem of Capelli (see [20] or p. 92 in [24]), λ is an algebraic integer of degree k. Thus the difference between two distinct numbers in the form of k−1 j=0 b j λ j is non-integer. We conclude that A G1 (w) = p(e 2πiw )(1 − e −2πiw ) h .…”

Section: Consider the Fourier Transformmentioning

confidence: 99%

Refinement equations and spline functions

Dubickas

2008

Adv Comput Math

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In this paper, we exploit the relation between the regularity of refinable functions with non-integer dilations and the distribution of powers of a fixed number modulo 1, and show the nonexistence of a non-trivial C ∞ solution of the refinement equation with non-integer dilations. Using this, we extend the results on the refinable splines with non-integer dilations and construct a counterexample to some conjecture concerning the refinable splines with non-integer dilations. Finally, we study the box splines satisfying the refinement equation with non-integer dilation and translations. Our study involves techniques from number theory and harmonic analysis.

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Section: Consider the Fourier Transformmentioning

confidence: 99%

Refinement equations and spline functions

Dubickas

2008

Adv Comput Math

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“…The usual criterion for the irreducibility of conics in the projective plane, in terms of the non-vanishing of the determinant of the associated quadratic form ( [11], p. 212), now applies to show that Q k is reducible if and only if k = 0, 1.…”

Section: Examplesmentioning

confidence: 99%

Finiteness of prescribed fibers of local biholomorphisms: a geometric approach

Chen

Xavier

2014

Math. Ann.

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Let X be a Stein manifold of complex dimension at least two, F : X → C n a local biholomorphism, and q ∈ F (X). In this paper we formulate sufficient conditions, involving only objects naturally associated to q, in order for the fiber over q to be finite. Assume that F −1 (l) is 1-connected for the generic complex line l containing q, and F −1 (l) has finitely many components whenever l is an exceptional line through q. Using arguments from topology and differential geometry, we establish a sharp estimate on the size of F −1 (q). It follows that for n ≥ 2 a local biholomorphism of X onto C n is invertible if and only if the pull-back of every complex line is 1-connected.

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“…Of special importance is Schinzel's book [12], which contains beautiful proofs using (in most cases) only basic properties of polynomials. In the first few sections of this paper, we include new proofs of some known results.…”

Section: α) and [K(α) : K] Is Divisible By A Prime Greater Than Max(mentioning

confidence: 99%