Degree spectra for transcendence in fields

We show that for both the unary relation of transcendence and the finitary relation of algebraic independence on a field, the degree spectra of these relations may consist of any single computably enumerable Turing degree, or of those c.e. degrees above an arbitrary fixed $\Delta^0_2$ degree. In other cases, these spectra may be characterized by the ability to enumerate an arbitrary $\Sigma^0_2$ set. This is the first proof that a computable field can fail to have a computable copy with a computable transcendence basis

АЛГОРИТМИЧЕСКАЯ НЕЗАВИСИМОСТЬ ЕСТЕСТВЕННЫХ ОТНОШЕНИЙ НА ВЫЧИСЛИМЫХ ЛИНЕЙНЫХ ПОРЯДКАХ // Ученые записки КФУ. Физико-математические науки 2013 том155 N3

Рассмотрены вопросы алгоритмической зависимости различных отношений на линейных порядках. Доказано, что отношения соседства, блока, плотности, предельности справа и предельности слева являются алгоритмически независимыми. Введены новые отношения, определимые в сигнатуре линейного порядка, являющиеся алгоритмически зависимыми, и изучены их свойства

On the complexity of the successivity relation in computable linear orderings

In this paper, we solve a long-standing open question (see, e.g., Downey [6, §7] and Downey and Moses [11]), about the spectrum of the successivity relation on a computable linear ordering. We show that if a computable linear ordering L has infinitely many successivities, then the spectrum of the successivity relation is closed upwards in the computably enumerable Turing degrees. To do this, we use a new method of constructing ∆ 0 3-isomorphisms, which has already found other applications such as Downey, Kastermans and Lempp [9] and is of independent interest. It would seem to promise many further applications