We show that deciding square-freeness of a sparse univariate
polynomial over the integer and over the algebraic closure of a
finite field is NP-hard. We also discuss some related open
problems about sparse polynomials.
We design polynomial size, constant depth (namely, $AC^0$) arithmetic formulae for the greatest common divisor (GCD) of two polynomials, as well as the related problems of the discriminant, resultant, Bézout coefficients, squarefree decomposition, and the inversion of structured matrices like Sylvester and Bézout matrices. Our GCD algorithm extends to any ... more >>>
