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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

A. Białynicki-Birula, A. Schinzel (2008)

Colloquium Mathematicae

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...

Displaying 1 – 10 of 10

Reducibility of a special symmetric form

Reducibility of lacunary polynomials II

Reducibility of lacunary polynomials. X

Reducibility of lacunary polynomials XII

Reducibility of Symmetric Polynomials

Reduction and specialization of polynomials

Representations of multivariate polynomials by sums of univariate polynomials in linear forms

Ritt's Second Theorem in arbitrary characteristic.

Root continuity theorems in valued fields.

Root continuity theorems in valued fields. II.

Currently displaying 1 – 10 of 10