Realization of finitely generated profinite groups by maximal abelian extensions of fields.
Realizing dyadic factors of elementary type Witt rings and pro-2 Galois groups.
Recent results in hyperring and hyperfield theory.
Recent results in the theory of constant reductions
Recent work on differential Galois theory
Reciprocity in valued function fields.
Reducibility of a special symmetric form
Irreducibility over of a special symmetric form in a variables is proved for .
Reducibility of lacunary polynomials II
Reducibility of lacunary polynomials, III
Reducibility of lacunary polynomials, V
Reducibility of lacunary polynomials. X
Reducibility of lacunary polynomials, XI
Reducibility of lacunary polynomials XII
Reducibility of quadrinomials
Reducibility of Symmetric Polynomials
A necessary and sufficient condition is given for reducibility of a symmetric polynomial whose number of variables is large in comparison to degree.
Reducible lacunary polynomials
Reducible Linear Difference Operators
Reduction and specialization of polynomials
We show explicit forms of the Bertini-Noether reduction theorem and of the Hilbert irreducibility theorem. Our approach recasts in a polynomial context the geometric Grothendieck good reduction criterion and the congruence approach to HIT for covers of the line. A notion of “bad primes” of a polynomial P ∈ ℚ[T,Y] irreducible over ℚ̅ is introduced, which plays a central and unifying role. For such a polynomial P, we deduce a new bound for the least integer t₀ ≥ 0 such that P(t₀,Y) is irreducible...
Reduction of differential equations
Let (F,D) be a differential field with the subfield of constants C (c ∈ C iff Dc=0). We consider linear differential equations (1) , where , and the solution y is in F or in some extension E of F (E ⊇ F). There always exists a (minimal, unique) extension E of F, where Ly=0 has a full system of linearly independent (over C) solutions; it is called the Picard-Vessiot extension of F E = PV(F,Ly=0). The Galois group G(E|F) of an extension field E ⊇ F consists of all differential automorphisms of...
Reduction of semialgebraic constructible functions
Let R be a real closed field with a real valuation v. A ℤ-valued semialgebraic function on Rⁿ is called algebraic if it can be written as the sign of a symmetric bilinear form over R[X₁,. .., Xₙ]. We show that the reduction of such a function with respect to v is again algebraic on the residue field. This implies a corresponding result for limits of algebraic functions in definable families.