Classifications of Real Conic and Cubic Curves

Polynomials in two variables with real-number coefficients of total degree at most three are considered. The goal is to address when two polynomials can be considered equivalent from the perspective of algebraic geometry. First, we say two polynomials are equivalent if an affine linear transformation of two-dimensional real space can bijectively map the solution set of one polynomial to the solution set of the other. Second, we will say two polynomials are equivalent if any automorphism of the polynomial ring in two variables with real coefficients can bijectively map the solution set of one polynomial to the solution set of the other. Third, we will say two polynomials are equivalent if their respective affine coordinate rings are ring isomorphic for an isomorphism that fixes the real numbers. We seek a sharp list of representatives for equivalence classes of polynomials with respect to each of the three equivalences mentioned. For the first two equivalences, a full solution is provided. A partial solution is included in the case of the third.