Ideals, Varieties, and Algorithms by David A. Cox John Little & Donal O’Shea

Author:David A. Cox, John Little & Donal O’Shea
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Thus, the conditions of Corollary 9 mean that g “follows generically over ” from the hypotheses .

This interpretation points out what is perhaps the main limitation of the Gröbner basis method in geometric theorem proving: it can only prove theorems where the conclusions follow generically over , even though we are only interested in what happens over . In particular, there are theorems which are true over but not over [see STURMFELS (1989) for an example]. Our methods will fail for such theorems.

When using Corollary 9, it is often unnecessary to consider the radical of . In many cases, the first power of the conclusion is in already. So most theorem proving programs in effect use an ideal membership algorithm first to test if , and only go on to the radical membership test if that initial step fails.

To illustrate this, we continue with the Circle Theorem of Apollonius from Example 3. Our hypotheses are the eight polynomials from (5)–(7). We begin by computing a Gröbner basis (using lex order) for the ideal , which yields

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