Uppers to zero in polynomial rings

2006 ◽  
pp. 243-261 ◽  
Author(s):  
Evan Houston
Keyword(s):  
Polynomial Rings

scholarly journals PI degree parity in q-skew polynomial rings

2008 ◽  
Vol 319 (10) ◽  
pp. 4199-4221 ◽  
Author(s):  
Heidi Haynal
Keyword(s):  
Polynomial Rings ◽  
Skew Polynomial Rings ◽  
Skew Polynomial

The Goldbach 3-primes property for polynomial rings over certain infinite fields

1998 ◽  
Vol 43 (15) ◽  
pp. 1256-1260 ◽  
Author(s):  
Shiqiang Wang
Keyword(s):  
Polynomial Rings

scholarly journals Down–Up Algebras and Ambiskew Polynomial Rings

2000 ◽  
Vol 228 (1) ◽  
pp. 311-346 ◽  
Author(s):  
David A Jordan
Keyword(s):  
Polynomial Rings

On the validity of hilbert's nullstellensatz, artin's theorem, and related results in grothendieck toposes

1988 ◽  
Vol 53 (4) ◽  
pp. 1177-1187
Author(s):  
W. A. MacCaull
Keyword(s):  
Intuitionistic Logic ◽  
Main Idea ◽  
Rational Functions ◽  
Completeness Theorem ◽  
Point Of View ◽  
Polynomial Rings ◽  
Von Neumann ◽  
Ordered Fields ◽  
Von Neumann Regular ◽  
Theoretic Point

Using formally intuitionistic logic coupled with infinitary logic and the completeness theorem for coherent logic, we establish the validity, in Grothendieck toposes, of a number of well-known, classically valid theorems about fields and ordered fields. Classically, these theorems have proofs by contradiction and most involve higher order notions. Here, the theorems are each given a first-order formulation, and this form of the theorem is then deduced using coherent or formally intuitionistic logic. This immediately implies their validity in arbitrary Grothendieck toposes. The main idea throughout is to use coherent theories and, whenever possible, find coherent formulations of formulas which then allow us to call upon the completeness theorem of coherent logic. In one place, the positive model-completeness of the relevant theory is used to find the necessary coherent formulas.The theorems here deal with polynomials or rational functions (in s indeterminates) over fields. A polynomial over a field can, of course, be represented by a finite string of field elements, and a rational function can be represented by a pair of strings of field elements. We chose the approach whereby results on polynomial rings are reduced to results about the base field, because the theory of polynomial rings in s indeterminates over fields, although coherent, is less desirable from a model-theoretic point of view. Ultimately we are interested in the models.This research was originally motivated by the works of Saracino and Weispfenning [SW], van den Dries [Dr], and Bunge [Bu], each of whom generalized some theorems from algebraic geometry or ordered fields to (commutative, von Neumann) regular rings (with unity).

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