For an odd positive integer n ≥ 5, assuming the truth of the abc conjecture, we show that for a positive proportion of pairs (a, b) of integers the trinomials of the form tn + at + b (a, b ∈ Z) are irreducible and their discriminants are squarefree.

An efficient algorithm for the multiplication in GF (2) was introduced by Mastrovito. The space complexity of the Mastrovito multiplier for the irreducible trinomial x +x+1 was given as m − 1 XOR and m AND gates. In this paper, we describe an architecture based on a new formulation of the multiplication matrix, and show that the Mastrovito multiplier for the generating trinomial x + x + 1, wher...

We prove that any pair of bivariate trinomials has at most 16 roots in the positive quadrant, assuming there are only finitely many roots there. The best previous upper bound independent of the polynomial degrees (following from a general result of Khovanski with stronger non-degeneracy hypotheses) was 248,832. Our proof allows real exponents and extends to certain systems of n-variate fewnomials.

We construct classes of permutation polynomials over FQ2 by exhibiting classes of low-degree rational functions over FQ2 which induce bijections on the set of (Q + 1)-th roots of unity. As a consequence, we prove two conjectures about permutation trinomials from a recent paper by Tu, Zeng, Hu and Li.

We give a new algorithm for performing the distinct-degree factorization of a polynomial P (x) over GF(2), using a multi-level blocking strategy. The coarsest level of blocking replaces GCD computations by multiplications, as suggested by Pollard (1975), von zur Gathen and Shoup (1992), and others. The novelty of our approach is that a finer level of blocking replaces multiplications by squarin...

We prove that any pair of bivariate trinomials has at most 5 isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees counted only non-degenerate roots and even then gave much larger bounds, e.g., 248832 via a famous general result of Khovanski. Our bound is sharp, allows real exponents, and extends to certain systems of n-variate fewnomials....