A Large Algebraically Closed Field

The Multiplicative Inverse Eigenvalue Problem over an Algebraically Closed Field

Let M be an n × n square matrix and let p(λ) be a monic polynomial of degree n. Let Z be a set of n × n matrices. The multiplicative inverse eigenvalue problem asks for the construction of a matrix Z ∈ Z such that the product matrix MZ has characteristic polynomial p(λ). In this paper we provide new necessary and sufficient conditions when Z is an affine variety over an algebraically closed field.

Pseudo Algebraically Closed Extensions

Decomposition of Homogeneous Polynomials over an Algebraically Closed Field

Let F be a homogeneous polynomial of degree d in m+ 1 variables defined over an algebraically closed field of characteristic zero and suppose that F belongs to the s-th secant varieties of the standard Veronese variety Xm,d ⊂ P( m+d d )−1 but that its minimal decomposition as sum of d-th powers of linear forms M1, . . . ,Mr is F = M 1 +· · ·+M r with r > s. We show that if s+r ≤ 2d+1 then such ...

A Note on Existentially Closed Difference Fields with Algebraically Closed Fixed Field

We point out that the theory of difference fields with algebraically closed fixed field has no model companion. By a difference field we mean a field K equipped with an automorphism σ. It is well-known ([?]) that the class of existentially closed difference fields is an elementary class (ACFA), and moreover all completions are unstable. The fixed field (set of a ∈ K such that σ(a) = a) is respo...

Strong boundedness and algebraically closed groups

Let G be a non-trivial algebraically closed group and X be a subset of G generating G in infinitely many steps. We give a construction of a binary tree associated with (G,X). Using this we show that if G is ω1-existentially closed then it is strongly bounded.