I am teaching myself about algebraic geometry, and the classification of singular points on algebraic curves $f(x,y)=0$, where $x,y\in\mathbb{C}$.
One way to classify these singular points (a set of $P$'s) is by looking at their multiplicities. It seems like there are two definitions:
Definition 1 (Partial Derivatives): The multiplicity, $m$, at $P$ can be defined as the smallest positive integer for which there exists some partial derivative of order $m$ that doesnt vanish at $P$.
Definition 2 (Ideal Membership): Another way to determine the multiplicity, particularly in algebraic geometry, is by looking at the ideal generated by $f$ and its partial derivatives, and then determining the dimension of the quotient space $\mathbb{C}[x,y]/I$, where $I$ is the ideal generated by $f$, $f_x$, and $f_y$. The dimension of this quotient space as a $\mathbb{C}$-vector space gives the multiplicity of $P$.
It seems like the second method is more general and can handle more complex cases as well.
I would like to know how to compute this quotient ideal in some computer algebra system. Maple is preferable, but other systems are welcome. An example calculation would be particularly useful either in computer algebra language, or by hand. Thanks!
If all you want is the dimension, you don't need to actually compute the quotient. Instead, I think the standard approach is to use Grobner bases. There is a result that says that you can calculate the dimension of $\mathbb{C}[x,y]/I$ by counting how many monomials are not contained in the ideal generated by the "leading monomials" of elements of $I$ under a monomial order. This is Theorem 15.3 in Eisenbud's commutative algebra book, and I'm sure there are plenty of other references for it as well. (Eisenbud works in more generality than most other sources, so if you want a good introduction to Grobner bases before turning to Eisenbud, I'd suggest Dummit and Foote, or else Cox, Little, and O'Shea's Ideals, Varieties, and Algorithms.)
I don't want to type up the theory of monomial orders and Grobner bases from scratch here, but I'll include a practical explanation of how to calculate the dimension using Macaulay2. Since things sometimes go haywire over $\mathbb{C}$, I'll work over $\mathbb{Q}$, although it really doesn't matter much for these calculations.
Say I want to find the dimension of $\mathbb{Q}[x,y]/I$ where $I=(2xy+y^2,2xy+x^2)$—these are the partials of $x^2y+xy^2$. I first have Macaulay2 calculate a Grobner basis with respect to whatever monomial order I want (here I'm not specifying one, so Macaulay2 just uses its default):
This outputs
What matters is the second line, where it lists out the Grobner basis. The dimension of the quotient is then the number of monomials not contained in the ideal generated by the first terms of the polynomials in the Grobner basis. (Macaulay2 automatically orders the terms properly so that the first terms are the "leading" terms with respect to the chosen monomial order.) Here those leading terms are $2xy$, $x^2$, and $y^3$. The monomials not contained in $(2xy,x^2,y^3)$ are $1$, $x$, $y$, and $y^2$. So Eisenbud Theorem 15.3 says that these monomials are a basis (over $\mathbb{Q}$) for $\mathbb{Q}[x,y]/I$, so the the dimension of $\mathbb{Q}[x,y]/I$ is $4$, which says that $x^2y+xy^2$ has a multiplicity-$4$ singularity at the origin.
(I'll also just mention that I think things go wrong if the function in question has more singularities not at the origin. To fix this, we instead consider the dimension of $k[x,y]_{(x,y)}/I$, localizing first at the origin. But then ordinary Grobner bases aren't as helpful. This is discussed Chapter 4 of Cox's Using Algebraic Geometry—look for the discussion of Milnor numbers. Extending the algorithm to this case involves replacing the ordinary monomial order usually used for Grobner bases with a "local" monomial order, which leads to so-called standard bases. These are also implemented in Macaulay2, although I haven't used them myself.)