18 research outputs found
Irreducibility Criteria for Polynomials with non-negative Coefficients
No full textIn [7, b.2, VIII, 128] Pólya and Szegö state the following theorem of A. Cohn:THEOREM 1. Let dndn−x … d0 be the decimal representation of a prime. Thenis irreducible.Thus, for example, since 1289 is prime, x3 + 2x2 + 8x + 9 is irreducible. Brillhart, Odlyzko, and the author generalized Cohn's Theorem in three different directions. As examples of these types of generalizations, we note the following results, the first two of which are special cases of a result in [1] and the third of a result in [3].THEOREM 2. Let dndn−x … d0 be the base b representation of a prime where b is an integer ≧2. Thenis irreducible.THEOREM 3. Letbe such that f(10) is prime and 0 ≦ dj ≦ 167 for j = 0, 1, …, n. Then f(x) is irreducible.</jats:p
Widely Digitally Delicate Brier Primes and Irreducibility Results for Some Classes of Polynomials
No full textThis dissertation considers three different sections of results. In the first part of the dissertation, a result on consecutive primes which are widely digitally delicate and Brier numbers is discussed. Making use of covering systems and a theorem of D. Shiu, M. Filaseta and J. Juillerat showed that for every positive integer k, there exist k consecutive widely digitally delicate primes. They also noted that for every positive integer k, there exist k consecutive primes which are Brier numbers. We show that for every positive integer k, there exist k consecutive primes that are both widely digitally delicate and Brier numbers. This is joint work with M. Filaseta and J. Juillerat.
In the second part of the dissertation, we prove an irreducibilty result for a class of polynomials. Consider the polynomial F(x) = f(x)+Mg(x) where M is a positive integer and f(x), g(x) ∈ Z[x] such that gcd(f(x), g(x)) = 1. A version of Hilbert’s Irreducibility Theorem in this setting implies that F(x) is irreducible for almost all M. In the case that deg f \u3c deg g, recent results by M. Cavachi, M. Vajaitu, and A. Zaharescu [14] and by N.C. Bonciocat, Y. Bugeaud, M. Cipu, and M. Mignotte [9] have given definitive examples where irreducibility occurs by taking M to be a prime power bounded below by an explicit function depending on f and g. We provide a wider class of definitive examples by taking M with a large prime factor, and in particular our explicit examples include a set of M with positive asymptotic density in the integers. We then extend the result to bivariate polynomials in a manner similar to work by N.C. Bonciocat, Y. Bugeaud, M. Cipu, and M. Mignotte [9]. This is joint work with M. Filaseta.
In the third part of the dissertation, we prove the irreducibility of nth order Euler polynomials of even degree n. For m an even positive integer and p a prime, we show that the generalized Euler polynomial
E(mp) mp (x) is in Eisenstein form with respect to p if and only if p does not divide m(2m − 1)Bm. As a consequence, we deduce that at least 1/3 of the generalized Euler polynomials E(n) n(x) are in Eisenstein form with respect to a prime p dividing n and, hence, irreducible over Q. This is joint work with M. Filaseta
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On the Galois Group of generalized Laguerre polynomials
No full textThis is the pre-published version harvested from ArXiv. The published version is located at http://jtnb.cedram.org/item?id=JTNB_2005__17_2_517_0
http://archive.numdam.org/ARCHIVE/JTNB/JTNB_2005__17_2/JTNB_2005__17_2_517_0/JTNB_2005__17_2_517_0.pdfUsing the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be “large.” For a fixed α∈ℚ-ℤ <0 , Filaseta and Lam have shown that the nth degree Generalized Laguerre Polynomial L n (α) (x)=∑ j=0 n n+α n-j(-x) j /j! is irreducible for all large enough n. We use our criterion to show that, under these conditions, the Galois group of L n (α) (x) is either the alternating or symmetric group on n letters, generalizing results of Schur for α=0,1,±1 2,-1-n.517-52
Irreducibility Criteria For Polynomials With Non-Negative Integer Coefficients, and the Prime Factorization of F(N) For F(X) In Z[X]
No full textWe explore two specific connections between prime numbers and polynomials. Cohn\u27s Criterion states that if is the base representation of a prime, then the polynomial is irreducible. Let be a polynomial with non-negative integer coefficients. We define be the largest integer such that if is prime and all the coefficients of are , then is irreducible. It is known that We improve the lower bound above to . Furthermore, we classify all reducible polynomials with non-negative integer coefficients such that is prime and all the coefficients of are . Let be a finite set of rational primes. For a non-zero integer , define % , where is the usual -adic norm of . In 1984, Stewart applied Baker\u27s theorem to prove non-trivial, computationally effective upper bounds for for any integer . Effective upper bounds have also been given by Bennett, Filaseta, and Trifonov for and , where and , respectively. We extend Stewart\u27s theorem to prove effective upper bounds for for an arbitrary in having at least two distinct roots
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Algebraic Properties of a Family of Generalized Laguerre Polynomials
No full textThis is the pre-published version harvested from ArXiv. The published version is located at http://www.math.ca/10.4153/CJM-2009-031-6We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers r,n≥0 , we conjecture that L(−1−n−r)n(x)=∑nj=0(n−j+rn−j)xj/j! is a \Q -irreducible polynomial whose Galois group contains the alternating group on n letters. That this is so for r=n was conjectured in the 1950's by Grosswald and proven recently by Filaseta and Trifonov. It follows from recent work of Hajir and Wong that the conjecture is true when r is large with respect to n≥5 . Here we verify it in three situations: i) when n is large with respect to r , ii) when r≤8 , and iii) when n≤4 . The main tool is the theory of p -adic Newton Polygons.583-60
Classes of polynomials having only one non-cyclotomic irreducible factor, preprint
No full textsecond author stated the following: Conjecture 1. Let n be an integer 2, and let f(x) =1+x+x 2 + +x n. Then f 0 (x) is irreducible over the rationals
On Galois groups of a one-parameter orthogonal family of polynomials
No full textFor a fixed integer t>1, we show that if t is not equal to 2, a square ≥4, or three times a square, then the discriminant of the generalized Laguerre polynomial L(s/t)n(x) is a nonzero square for at most finitely many pairs (n,s). Otherwise, the discriminant of L(s/t)n(x) is a nonzero square if and only if (n,s) belongs to one of seven explicitly describable infinite sets or to an additional finite set. This extends the results obtained for t=1 by P. Banerjee, M. Filaseta, C. Finch and J. Leidy. As a consequence, if α is a fixed rational number not equal to 1, 3, 5, or a negative integer, then for all but finitely many n, L(α)n(x) has Galois group Sn, thereby refining a previous result of M. Filaseta – T. Y. Lam and F. Hajir. As an illustration, we give for t=2 infinitely many integer specializations (n,s(n)) such that L(s(n)/2)n(x) has Galois group An. For n≤5, the set of rational numbers α for which the discriminant of L(α)n(x) is a nonzero square is explicitly computed by solving certain generalized Pell-like equations
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Neutron Production of Charm Particles in Fermilab E-400
No full textResults are presented from Fermilab E-400 on the production of charmed baryons and mesons using incident neutrons. We show evidence for the charm-strange baryon, ..xi../sub c//sup +/, and present our measurements of its mass, width, lifetime, cross section and relative branching fractions, and the A, x/sub f/, p/sub t/, and particle/antiparticle dependence of the state. We show evidence for both the ..sigma../sub c//sup + +/ and ..sigma../sub c//sup 0/, and present measurements of three mass differences, ..sigma../sub c//sup + +/ - ..sigma../sub c//sup 0/, ..sigma../sub c//sup 0/ - ..lambda../sub c//sup +/, and ..sigma../sub c//sup + +/ - ..lambda../sub c//sup +/. Preliminary results on the ratio of two decay modes of the D/sup 0/ are shown. D/sup 0/ ..-->.. K/sup +/K/sup -/ and D/sup 0/ ..-->.. K/sub 0/ anti K/sub 0/. The latter mode has not been previously observed. 8 refs., 10 figs
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Charm Hadroproduction Results From Fermilab E-400
No full textResults are presented from Fermilab E-400 on the production of charmed baryons and mesons at a mean energy of 640 GeV. We show evidence for the charm-strange baryon, ..xi../sub c//sup +/, and present our measurements of its mass, width, lifetime, cross section and relative branching fractions, and the A, x/sub f/, p/sub t/, and particle/antiparticle dependence of the state. We show evidence for both the ..sigma../sub c//sup 2 +/ and ..sigma../sub c//sup 0/, and present measurements of three mass differences, ..sigma../sub c//sup 2 +/ - ..sigma../sub c//sup 0/, ..sigma../sub c//sup 0/ - ..lambda../sub c//sup +/, and ..sigma../sub c//sup 2 +/ - ..lambda../sub c//sup +/. Measurements of the A dependence and particle/antiparticle ratios for ..sigma../sub c/ production are also presented. We show preliminary results on the ratio of two decay modes of the D/sup 0/, D/sup 0/ ..-->.. K/sup +/K/sup -/ and D/sup 0/ ..-->.. K/sub 0/anti K/sub 0/. The latter mode has not been previously observed. 8 refs., 10 figs
Measurement of the form factors for the decay D(s)+ → phi mu+ nu
No full textThe fermilab high-energy photoproduction experiment E687 provides a sample of approximately 90 events of the decay mode D(s)+→ phi mu+ nu. The ratios of the form factors governing the decay are measured to be Rv=1.8+-0.9+-0.2 and R2= 1.1+-0.8+-0.1, implying a polarization of G1/Gt=1.0+-0.5+-0.1 for the electron decay, consistent with our measurement of the form factor for the decay D+->anti(K0*) mu+ nu
