By Walker R. J.

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**Extra resources for Algebraic Curves**

**Example text**

2. 3 to get algorithms for operations with polynomial ideals. 1. Intersection of ideals. In order to get a procedure to compute a Gr¨obner basis for the intersection of two polynomial ideals we ﬁrst need to introduce a notation. For an ideal I of S and a polynomial p(t) in a new variable t, we denote by p(t)I the ideal of S[t] which is generated by {p(t)f : f ∈ I} ⊂ S[t]. It is straightforward to notice that if I is generated by f1 , . . , fr as an ideal of S, then p(t)I is generated by p(t)f1 , .

We will now derive a criterion which allows us to answer this question in a ﬁnite number of steps. To explain this criterion we have to introduce the socalled S-polynomials. Suppose ﬁrst we are dealing with an ideal generated by two nonzero polynomials, say I = (f, g), and we want to compute a Gr¨obner basis of I. Certainly in< (f ) and in< (g) belong to in< (I). A candidate of a polynomial h ∈ I whose initial monomial does not belong to (in< (f ), in< (g)) is a linear combination of f and g such that their initial terms cancel.

N, and let < be the lexicographic order on S. Show that op

### Algebraic Curves by Walker R. J.

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