Our professor in algebra geometry wrote us the following in class deg(xy=0)=2. I don't understand what that is, I know the definition of degree for the following let p be a polynomial deg(p)=max(sum of the exponents of the variables of each monomial).But in my case I don't have a polynomial I have the set of solutions of an equation, what is the definition of degree in that case?
2026-05-16 18:25:22.1778955922
What is deg(xy=0)?
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Geometrically, $xy=0$ corresponds to the union of the coordinate axes in $\mathbb A^2$. Now, you can define the degree of one-dimensional varieties as the number of intersections with a generic line in $\mathbb A^2$. If you intersect $xy=0$ with a generic line, then you get two intersection points, so the degree should be two.
(But beware, this is really kinda false: $\mathbb A^2$ is not compact, so intersection theory does not work nicely. In some sense, everything should have degree zero in $\mathbb A^2$, because you can "slide divisors away to infinity".)
Algebraically, the degree of any curve in $\mathbb A^2$ is the degree of its maximal term.