Introduction to algebraic geometry

Let $\mathbb{K}$ be a fixed body (in this article, the $\mathbb{Q}$ body of rational numbers, the $ℝ$ body of real numbers or the $ℂ$ body of complex numbers).

Consider a system of equations P _i (X ₁ ,..., X _n ) = 0 where the P _i are polynomials with coefficients in $\mathbb{K}$ and the variables take their values in $\mathbb{K}$ (or a body containing $\mathbb{K}$ ). Typically, the equations will have integer coefficients, and we will consider the solutions of the system in ${ℝ}^n$ or ${ℂ}^n\mathrm{.}$

Such a system defines a subset $\mathrm{X}\subset {\mathbb{K}}^n$ called an algebraic variety, and algebraic geometry consists in understanding the "geometric" properties of X using the "algebraic" properties of the system of equations. A fundamental historical example is the study of the trajectories of planets and comets, which are, to a first approximation, plane curves of the second degree (i.e. conics: ellipses, parabolas or hyperbolas).

Although in practice we're interested (as in the example above) in the real case, we find that the relationship sought between...