Osculating Curves
Abstract
Consider a complex analytic curve $X$ in $\mathbb{C}^2$, along with a specific point $p \in X$. The primary concern arises in approximating geometrically the curve $X$ precisely at the point $p$. Analogously, in introductory calculus, students learn to compute the tangent line to the graph of a function $y=f(x)$ at a given point $p=(x_*,f(x_*))$ by utilizing the derivative of $f$ at $x_*$.
For analytical convenience, we assume a local representation of the curve $X$ using a power series expansion. This representation centers the point $p$ at the origin $(0,0) \in \mathbb{C}^2$. Thus, our mathematical input becomes a Taylor series:
\[ f(x) = \sum_{i=1}^\infty c_ix^i \ . \]
Alternatively, one may conceptualize our input as a ``germ'' or a ``jet'' of sufficiently high order. This assumption of a local representation is grounded in the application of the implicit function theorem. The theorem, coupled with the specific nature of the questions posed, allows us to work with a localized description of $X$ that relies on only finitely many coefficients $\{c_i\}_{i=1}^\infty$.
An alternative perspective on defining a plane curve involves considering it as the set of points where an implicit polynomial function $F(x,y) \in \mathbb{C}[x,y]$ equals zero. This set of points is termed a ``variety,'' denoted as $\mathcal V(F(x,y)) = \{(x,y) \in \mathbb{C}^2 \mid F(x,y)=0\}$.
Given a degree $d$ and a sequence $\textbf{c}=(c_1,c_2,\ldots)$ defining the power series expansion, our objective is to ascertain the ``best approximation'' of the curve $\Gamma$ through a curve expressed as $\mathcal V(F(x,y))$, where $F(x,y)$ has a degree no greater than $d$. This specific curve is referred to as the ``degree $d$ osculating curve'' of the Taylor series, or equivalently, the ``degree $d$ osculating curve'' of $X$ at the point $p$.