Reducibility without qualification means in this lecture reducibility over the rational field. Questions on such reducibility occupy an intermediate place between questions on reducibility of polynomials over an algebraically closed field and those on primality. I shall refer to these two cases as to the algebraic and the arithmetic one and I shall try to exhibit some of the analogies consideri...

This paper deals with a q-analog of Painlevé III equation of type D (1) 7 . We study its algebraic function solutions and transcendental function solutions. We construct algebraic function solutions expressed by Laurent polynomials and prove irreducibility in the sense of decomposable extensions.

Two results in Differential Algebra, Kolchin’s Irreducibility Theorem, and a result on descent of projective varieties (due to Buium) are proved using methods of “modern” or “Grothendieck style” algebraic geometry.

We study the differential properties of generalized arc schemes and geometric versions of Kolchin’s Irreducibility Theorem over arbitrary base fields. As an intermediate step, we prove an approximation result for arcs by algebraic curves.

We study the Irreducibility Problem for arc space X∞ of an irreducible singular algebraic variety X defined over a perfect field k characteristic p>0. The existence reducible such is related to fact that Kolchin's Theorem fails hold in positive characteristic. obtain complete description components when surface. Section 5 introduces two main new problems arbitrary dimension: (1) line O. Zariski...

Using recent absolute irreducibility testing algorithms, we derive new irreducibility forms. These are integer polynomials in variables which are the generic coefficients of a multivariate polynomial of a given degree. A (multivariate) polynomial over a specific field is said to be absolutely irreducible if it is irreducible over the algebraic closure of its coefficient field. A specific polyno...

Abstract Let C be a complex projective smooth curve and W symplectic vector bundle of rank 2 n over . The Lagrangian Quot scheme $$LQ_{-e}(W)$$ L Q - e ( W ) para...