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Wikipedia says

There is a unique function ... $I_{p}(P,Q)$ called the intersection multiplicity of $P$ and $Q$ at $p$ that satisfies the following [six] properties:

...

Although these properties completely characterize intersection multiplicity, in practice it is realised in several different ways.

and then goes on to list a few realizations:

  1. the dimension of $K[[x, y]]/(P, Q)$ (assuming $p = (0, 0)$)
  2. the highest power of $y$ dividing the resultant of $P$ and $Q$
  3. using pertubations of the curves

Other sources (e.g., Fulton) also define it as the dimension of a quotient of the local ring at $p$.

However, why not define it purely computationally? It seems simple enough to enumerate all the cases where the degrees of $P$ and $Q$ are $\le 1$, and then define a procedure where you can reduce the degree of whichever of $P$ and $Q$ has the highest-degree term, using the given properties.

Are there textbooks that take such a computational approach? It's probably most suitable for a more introductory course, but it would avoid having to depend on a bunch of commutative algebra.

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  • $\begingroup$ The definition by the resultant is computational, isn't it? You could also have a look at p. 74 of Fulton, Algebraic Curves. $\endgroup$
    – Damian Rössler
    Commented Dec 8, 2023 at 15:14
  • $\begingroup$ Hmm, I don't see anything on p. 74 (dept.math.lsa.umich.edu/~wfulton/CurveBook.pdf#page=82). p. 37 does outline a computational method, but that's in the proof of uniqueness. For the proof of existence, Fulton seems to define it using the local ring at the point. I guess I'm wondering why the computational method isn't enough, and he still feels the need to prove existence using the local ring. $\endgroup$
    – Fred Akalin
    Commented Dec 8, 2023 at 18:36
  • $\begingroup$ "and then goes on to list a few realizations: 1. the dimension of $K[[x, y]]/(P, Q)$ (assuming $p = (0, 0)$)" - this property is "computational" in the sense that it defines the intersection multiplicity in terms of a "formula" involving the ideal $(P,Q)$. The exercise you must do is to understand how to calculate this dimension and how to prove these definitions are equivalent. $\endgroup$
    – hm2020
    Commented Jan 23 at 9:42

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