Is a conic a hypersurface defined by a degree 2 polynomial?
A conic is an intersection of a cone and a plane. So as a zero set, it is defined by a degree 2 polynomial and a linear equation. Now suppose this definition of a conic is equivalent to being a hypersurface defined by a degree 2 polynomial. Then it follows that the radical of the ideal say $(X^2 + Y^2 - Z^2, aX + bY + cZ)$ and the radical of our principal ideal generated by a degree 2 polynomial are equal. That is, the radical of our principal ideal contains a linear polynomial so is in fact a principal ideal generated by a linear polynomial...which is in fact a plane. A conic is definitely not a plane. What is my misunderstanding here?