Sparse univariate polynomials with many roots over finite fields.

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On means of positive definite matrices

AbstractIf f is a positive function on (0, ∞) which is monotone of order n for every n in the sense of Löwner and if Φ1 and Φ2 are concave maps among positive definite matrices, then the following map involving tensor products: (A,B)↦f[Φ1(A)−1⊗Φ2(B)]·(Φ1(A)⊗I) is proved to be concave. If Φ1 is affine, it is proved without use of positivity that the map (A,B)↦f[Φ1(A)⊗Φ2(B)−1]·(Φ1(A)⊗I) is convex. These yield the concavity of the map (A,B)↦A1−p⊗Bp (0<p⩽1) (Lieb's theorem) and the convexity of the map (A,B)↦A1+p⊗B−p (0<p⩽1), as well as the convexity of the map (A,B)↦(A·log[A])⊗I−A⊗log[B].These concavity and convexity theorems are then applied to obtain unusual estimates, from above and below, for Hadamard products of positive definite matrices