A Note on Polynomial Functions (original) (raw)
[Victor V. Prasolov] Polynomials (Algorithms and C)
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The Russian style of formulating mathematical problems means that nobody will be able to simplify your formulation as opposed to the French style which means that nobody will be able to generalize it,-Vladimir Arnold.
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2011
We show that the there exists an infinite family of APN functions of the form F (x) = x 2 s +1 + x 2 k+s +2 k + cx 2 k+s +1 + c 2 k x 2 k +2 s + δx 2 k +1 , over F 2 2k , where k is an even integer and gcd(2k, s) = 1, 3 ∤ k. This is actually a proposed APN family of Lilya Budaghyan and Claude Carlet who show in [6] that the function is APN when there exists c such that the polynomial y 2 s +1 + cy 2 s + c 2 k y + 1 = 0 has no solutions in the field F 2 2k. In [6] they demonstrate by computer that such elements c can be found over many fields, particularly when the degree of the field is not divisible by 3. We show that such c exists when k is even and 3 ∤ k (and demonstrate why the k odd case only re-describes an existing family of APN functions). The form of these coefficients is given so that we may write the infinite family of APN functions. APN functions; zeros of polynomials; irreducible polynomials.
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On a Class of Quadratic Polynomials with no Zeros and its Application to APN Functions
Eprint Arxiv 1110 3177, 2011
We show that the there exists an infinite family of APN functions of the form F(x)=x2s+1+x2k+s+2k+cx2k+s+1+c2kx2k+2s+deltax2k+1F(x)=x^{2^{s}+1} + x^{2^{k+s}+2^k} + cx^{2^{k+s}+1} + c^{2^k}x^{2^k + 2^s} + \delta x^{2^{k}+1}F(x)=x2s+1+x2k+s+2k+cx2k+s+1+c2kx2k+2s+deltax2k+1, over gf22k\gf_{2^{2k}}gf22k, where kkk is an even integer and gcd(2k,s)=1,3nmidk\gcd(2k,s)=1, 3\nmid kgcd(2k,s)=1,3nmidk. This is actually a proposed APN family of Lilya Budaghyan and Claude Carlet who show in \cite{carlet-1} that the function is APN when there exists ccc such that the polynomial y2s+1+cy2s+c2ky+1=0y^{2^s+1}+cy^{2^s}+c^{2^k}y+1=0y2s+1+cy2s+c2ky+1=0 has no solutions in the field gf22k\gf_{2^{2k}}gf22k. In \cite{carlet-1} they demonstrate by computer that such elements ccc can be found over many fields, particularly when the degree of the field is not divisible by 3. We show that such ccc exists when kkk is even and 3nmidk3\nmid k3nmidk (and demonstrate why the kkk odd case only re-describes an existing family of APN functions). The form of these coefficients is given so that we may write the infinite family of APN functions.
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Advances in Applied Mathematics, 2014
Rodrigues formulas Askey-Wilson operator Wilson operator Jackson q-difference operator Big and small q-Jacobi polynomial Al-Salam-Chihara polynomials Indeterminate moment problems We give new derivations of properties of the functions of the second kind of the Jacobi, little and big q-Jacobi polynomials, and the symmetric Al-Salam-Chihara polynomials for q > 1. We also study the Askey-Wilson functions and the Wilson functions of second kind. An integration by parts formula is derived for the Wilson operator in appropriate Hilbert space. In case of undetermined moment problem, we prove that the function of second kind depends on the measure.