Proof of the existence of certain triples of polynomials
Propriétés algébriques des polynômes en D.
Quadratic factors of f(x) -g(y)
Quasi-permutation polynomials
A quasi-permutation polynomial is a polynomial which is a bijection from one subset of a finite field onto another with the same number of elements. This is a natural generalization of the familiar permutation polynomials. Basic properties of quasi-permutation polynomials are derived. General criteria for a quasi-permutation polynomial extending the well-known Hermite's criterion for permutation polynomials as well as a number of other criteria depending on the permuted domain and range are established....
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. X
Reducibility of lacunary polynomials XII
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.
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...
Representations of multivariate polynomials by sums of univariate polynomials in linear forms
The paper is concentrated on two issues: presentation of a multivariate polynomial over a field K, not necessarily algebraically closed, as a sum of univariate polynomials in linear forms defined over K, and presentation of a form, in particular a zero form, as the sum of powers of linear forms projectively distinct defined over an algebraically closed field. An upper bound on the number of summands in presentations of all (not only generic) polynomials and forms of a given number of variables and...
Ritt's Second Theorem in arbitrary characteristic.
Root continuity theorems in valued fields.
Root continuity theorems in valued fields. II.
Salomon's Theorem for polynomials with several parameters
Schinzel's problem: Imprimitive covers and the monodromy method
Solved and unsolved problems οn polynomials
Some Algebraic Properties of Polynomial Rings
In this article we extend the algebraic theory of polynomial rings, formalized in Mizar [1], based on [2], [3]. After introducing constant and monic polynomials we present the canonical embedding of R into R[X] and deal with both unit and irreducible elements. We also define polynomial GCDs and show that for fields F and irreducible polynomials p the field F[X]/ is isomorphic to the field of polynomials with degree smaller than the one of p.
Some classes of irreducible polynomials
Specialization of quadratic and symmetric, bilinear forms, and a norm theorem