Algorithms and Bounds for Polynomial Rings

This chapter discusses algorithms and bounds for polynomial rings. A lemma discussed in the chapter combines with the well-known techniques and results of commutative algebra—local–global principles, Krull's intersection theorem, and the primitive element theorem—to obtain many of the bounds. The chapter describes the nonstandard approach. The results on bounds in their nonstandard formulation express very simple relations between two rings, a polynomial ring K[X] and a certain extension K[X]*. The lemma implies that a number A can be computed from (n,d) such that if f ∞ (f₁,…f_κ), then f = ∑h_if_i for certain h_i ∞ K[X] of degree at most A.