# Ideals, Varieties, and Algorithms: An Introduction to

Donal O'Shea
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To solve, or at least set up the language for a solution to an equivalence problem frequently involves understanding the functions deﬁned on an object. For each pair of hyperbolas. = 3( + 2) =1 = +2 2 ( − 2). = 2 2 − 1). 2 ( 2 − 2 − 1) 2 − 9 2 − 1) (16 2 − 9( + 2)2 − 1) Solution. ( 2 − 2 − 1) (8 − 1). (2) = 4 .5.2.. which has one connected component. to a hyperbola. (16 + 9( + 2)2 − 1) (4 2 + 2 − 6 + 8). (1) (2) (3) (4) ( 2 + 2 (( − 1) + − 1). 15 = = + + + + -plane to the -plane and - are two real aﬃne changes of coordinates from the from the -plane to the -plane. ﬁnd a real aﬃne change of coordinates that maps the hyperbola in the -plane to the hyperbola in the -plane. (4 2 + 9 2 − 1) (1) =4 .6.
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