Polynomial-Time Presentations of Algebraic Number Fields

Bit Complexity of Computing Solutions for Symmetric Hyperbolic Systems of PDEs (Extended Abstract)

“…The problem is to find a list of (codes of) all roots of (10) from given b(α 1 ) * · · · * b(α k ) and b(t 0 (x)) * · · · * b(t e (x)) where b is a natural binary encoding of C alg induced by the presentation of A described above and by Gauss representation of complex numbers as pairs of reals. As shown in [AS18], the problem is solvable in polynomial time for any fixed k. Moreover, the same estimate holds for the version of this problem when one computes the list of all distinct real roots of (10) in increasing order.…”

“…, k n ) for each n ≥ 1 which provide p-presentations of the corresponding rings and fields (see e.g. [AS18] for additionl details). Moreover, in the field Q(x 1 , .…”

“…If p ∈ Q[x] is unitary irreducible then the quotient-field Q[x]/(p) (formed by the poly-nomials of degree < deg(p)) also has a natural p-presentation (see e.g. [Ak89,AS18]). Therefore, the ordered field Q(α), for each non-zero α ∈ A, has a natural p-presentation.…”

Foundations of Online Structure Theory II: The Operator Approach

“…Also, as we see in the present paper, an 2 Although the definition above is not restricted to finite languages, we will never consider infinite languages in the paper. 3 Recall that a complete, first-order theory in a computable languadge is decidable if the collection of all Gödel codes of its sentences forms a computable set. A model upon the domain N is decidable if there is a Turing computable algorithm which, given a first-order formula with parameters from the domain of the structure, can decide whether the formula holds in the structure.…”

“…Also it would seem that although we can incorporate automatic processes in our theories, they are really not general enough for online algorithms in general. Similarly, polynomial time structures such as Cenzer and Remmel, Grigorieff, Alaev, and others [1,2,3,25,47] are rather presentation dependent. Finally, primitive recursive has a nice Church-Turing thesis, in that it models computable processes without unbounded loops.…”