I have a polynomial in multiple variables, and would like to have a transformation act on them so that the value of the polynomial is preserved. For example, the polynomial $x^2+y^2$ is preserved in rotation about the origin. Is there a general way to obtain transformations that maintain the values of polynomials? Do these transformations always exist?
I am looking for specifically those transformations that form a nice, smooth, continuous group (just like rotation or the Lorentz transformation), and would like sufficiently many of them that any point may be transformed to any other point on its level set by their application.
when the function is a quadratic form, meaning that, when we write the variables in a column vector $x,$ we have a symmetric matrix $H$ where the function is $x^T H x,$ the automorphism group is matrices $A$ such that $$ A^T HA = H $$ It is typical to demand $\det H \neq 0,$ which gives $\det A = \pm 1.$