1. Warming up to Enumerative Geometry 3 can certainly be written in the form a0x + a1. There are two cases to consider. In the first case, a1 = 0, we see that f(x) has no roots. The root can be viewed as having “gone off to ∞” as before: we can write f(x) = lim a0→0 a0x + a1, and then the root −a1/a0 goes to ∞ or −∞ as a0 approaches 0 from one direction or the other. If a1 = 0 as well, then there are infinitely many solutions: any number x trivially satisfies the equation 0·x = 0. The situation already gets much richer if d = 2. The quadratic formula gives x = −a1 ± a2 1 − 4a0a2 2a0 . There are now several possibilities: (1) a0 = 0. There are several well-known subcases. Let D = a1 2 − 4a0a2 be the discriminant of f(x). (a) D 0. There are two roots. (b) D 0. There are no real roots. (c) D = 0. There is one root. (2) a0 = 0. There are several possibilities: (a) a1 = 0. In this case, there are no solutions (unless additionally a2 = 0, in which case there are infinitely many roots). (b) a1 = 0. There is now exactly one solution x = −a2/a1. We can think that the “other root” has “gone off to infinity”. See Exercise 1. The situation will clearly only get worse as d increases. We can simplify the answer by changing the question. Essentially all of these cases can be unified neatly by a few simple changes in the enumerative problem: • Use complex coeﬃcients and solutions. • Count solutions with multiplicity. • Include infinity.

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