Nonstandard Arithmetic of Polynomial Rings

Nonstandard Methods for Bounds in Differential Polynomial Rings

Motivated by the problem of the existence of bounds on degrees and orders in checking primality of radical (partial) differential ideals, the nonstandard methods of van den Dries and Schmidt [“Bounds in the theory of polynomial rings over fields. A nonstandard approach.”, Inventionnes Mathematicae, 76:77–91, 1984] are here extended to differential polynomial rings over differential fields. Amon...

Differential Polynomial Rings of Triangular Matrix Rings

Nonstandard arithmetic and reverse mathematics

We show that each of the five basic theories of second order arithmetic that play a central role in reverse mathematics has a natural counterpart in the language of nonstandard arithmetic. In the earlier paper [HKK1984] we introduced saturation principles in nonstandard arithmetic which are equivalent in strength to strong choice axioms in second order arithmetic. This paper studies principles ...

Nonstandard arithmetic and recursive comprehension

First order reasoning about hyperintegers can prove things about sets of integers. In the author’s paper Nonstandard Arithmetic and Reverse Mathematics, Bulletin of Symbolic Logic 12 (2006), it was shown that each of the “big five” theories in reverse mathematics, including the base theory RCA0, has a natural nonstandard counterpart. But the counterpart RCA0 of RCA0 has a defect: it does not im...

Arithmetic Nullstellensatz and Nonstandard Methods

In this study we find height bounds for polynomial rings over integral domains. We apply nonstandard methods and hence our constants will be ineffective. Furthermore we consider unique factorization domains and possible bounds for valuation rings and arithmetical functions.