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ABSTRACT
This article is an introduction to algebraic geometry and some of its applications. Reminders of commutative algebra (groups, rings, ideals, ring fields of real and complex numbers) serve as an introduction. This article commences with the study of the roots of one variable polynomials, resultant, discriminant, Sturm sequences and real root isolation algorithms. Algebraic geometry in the affine and the projective plane (real and complex cases), singular points, Riemann surfaces and Harnack theorem are then dealt with. To conclude, the principle of certain elimination algorithms (intersection of two real plane curves, determination of the intrinsic equation of a parametric plane curve) is provided.
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Read the articleAUTHOR
-
Jean-Jacques RISLER: Professor emeritus - Pierre and Marie Curie University
INTRODUCTION
Let
be a fixed body (in this article, the
body of rational numbers, the
body of real numbers or the
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
and the variables take their values in
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KEYWORDS
real roots | resultant | Sturm sequences | algebraic plane curves | projective plane | Harnack's theorem | Riemann surfaces
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Mathematics
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Introduction to algebraic geometry
A little algebra
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