B. Dayton: The Functorial Property of the Global Dual

posted Mar 29, 2012, 6:24 AM by Tony Shaska
For an affine ring $R=\C[x_1,\dots,x_n]/I $ the global dual is the linear space $G(R)=\{d: R \rightarrow \C, | d \text{ is linear }\}$ where $\C$ is the field of complex numbers. If $\phi$ is a polynomial map $A^n \rightarrow A^m$ with $V(J)$ the closure of the image of $\phi(V(I))$ then there is a linear map $\phi_*:G(R) \rightarrow G(C[x_1,\dots,x_m]/J)$ making $\phi_*$ into a covariant functor from algebraic sets to $\C$-vector spaces. This can be implemented in terms of Sylvester and Macaulay matrices by a single matrix multiplication and easy extraction of an implicit description for $\overline{\phi(V(I))}$. Applications include implicitization of polynomial or rational parametric curves and surfaces, and affine or projective transformations of curves and surfaces. One can work numerically giving a nice method for computer graphing of space curves given implicitly.