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Bounds in the theory of polynomial rings over fields. A nonstandard approach. (English) Zbl 0539.13011
In this paper nonstandard methods are used to show the existence of bounds for polynomial rings over fields of which the following are typical: Let $$f,f_ 1,...,f_ m\in K[X_ 1,...,X_ n]=K[X]$$ of degree $$\leq d$$, K a field. (a) $$f\in(f_ 1,...,f_ m)$$ iff $$f=\Sigma f_ ih_ i$$ for certain $$h_ i\in K[X]$$ of degree $$\leq A$$; (b) $$(f_ 1,...,f_ m)$$ is prime or contains 1 iff $gh\in(f_ 1,...,f_ m)\Rightarrow g\in(f_ 1,...,f_ m)\quad or\quad h\in(f_ 1,...,f_ m)$ holds for all g, $$h\in K[X]$$ of degree $$\leq P$$. - Here A and P are bounds depending only on (n,d) (not on K, m, or the polynomials in question).
The results are proved by considering the ring extension $$K[X]\subset K[X]_{int}$$, K an internal field in some nonstandard structure, $$K[X]_{int}$$ the ring of internal polynomials in $$X_ 1,...,X_ n$$ over K. (a) follows from the faithful flatness of this extension, (b) from the results that prime ideals of K[X] remain prime when extended to $$K[X]_{int}$$. - By concentrating on the existence of bounds (rather than their construction) many complications of the original constructive approach are avoided.

##### MSC:
 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 12E05 Polynomials in general fields (irreducibility, etc.) 03H99 Nonstandard models 13B25 Polynomials over commutative rings
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