On the Jacobian conjecture and the configurations of roots.
On the Mulitplicity of Zeros of Polynomials over Arbitrary Finite Dimensional K-Algebras.
On the number of elliptic curves with CM cover large algebraic fields
Elliptic curves with CM unveil a new phenomenon in the theory of large algebraic fields. Rather than drawing a line between and or and they give an example where the line goes beween and and another one where the line goes between and .
On the number of real roots of a solvable polynomial
On the number of terms of a composite polynomial
On the number of terms of a power of a polynomial
On the number of zero trace elements in polynomial bases for F2n.
Let Fq denote the finite field of q elements. O. Ahmadi and A. Menezes have recently considered the question about the possible number of elements with zero trace in polynomial bases of F2n over F2. Here we show that the Weil bound implies that there is such a basis with n + O(log n) zero-trace elements.
On the reducibility type of trinomials
On the representation of cyclotomic polynomials as sums of squares
On the roots of polynomials over division algebras.
On the set of distances between two sets over finite fields.
On the zeros of polynomials over arbitrary finite dimensional algebras.
On trace forms of étale algebras and field extensions.
On trinominals of type x...+Ax...+1.
On varieties of Hilbert type
A variety over a field is of Hilbert type if is not thin. We prove that if is a dominant morphism of -varieties and both and all fibers , , are of Hilbert type, then so is . We apply this to answer a question of Serre on products of varieties and to generalize a result of Colliot-Thélène and Sansuc on algebraic groups.
Orthogonality and complementation in the lattice of subspaces of a finite vector space
We investigate the lattice of subspaces of an -dimensional vector space over a finite field with a prime power together with the unary operation of orthogonality. It is well-known that this lattice is modular and that the orthogonality is an antitone involution. The lattice satisfies the chain condition and we determine the number of covers of its elements, especially the number of its atoms. We characterize when orthogonality is a complementation and hence when is orthomodular. For...
Ostrwoski's theorem for Henselian valued skew fields.
PAC fields over number fields
We prove that if is a number field and is a Galois extension of which is not algebraically closed, then is not PAC over .
Periodic points, multiplicities, and dynamical units.
Pfister's Dimension and the Level of Fields.