Download e-book for iPad: Conics and Cubics: A Concrete Introduction to Algebraic by Robert Bix

Algebraic Geometry

By Robert Bix

Conics and Cubics is an obtainable advent to algebraic curves. Its specialise in curves of measure at such a lot 3 retains effects tangible and proofs obvious. Theorems persist with clearly from highschool algebra and key rules: homogenous coordinates and intersection multiplicities.

By classifying irreducible cubics over the true numbers and proving that their issues shape Abelian teams, the ebook supplies readers easy accessibility to the learn of elliptic curves. It encompasses a uncomplicated facts of Bezout's Theorem at the variety of intersections of 2 curves.

The booklet is a textual content for a one-semester direction on algebraic curves for junior-senior arithmetic majors. the single prerequisite is first-year calculus.

The new version introduces the deeper examine of curves via parametrization by way of strength sequence. makes use of of parametrizations are provided: counting a number of intersections of curves and proving the duality of curves and their envelopes.

About the 1st edition:

"The book...belongs within the admirable culture of laying the principles of a tough and almost certainly summary topic via concrete and available examples."

- Peter Giblin, MathSciNet

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Extra resources for Conics and Cubics: A Concrete Introduction to Algebraic Curves (Undergraduate Texts in Mathematics)

Example text

Exercises 47 (e) Following the transformation in (32) with that in (33). (f ) Following the transformation in (33) with that in (32). 10. Let A, B, C be three collinear points, and let A0 , B 0 , C 0 be three collinear points. 4 to prove that there is a transformation that maps A, B, C to A0 , B 0 , C 0 , respectively. 11. (a) Consider any transformation that fixes the origin, the point (1, 0) in the Euclidean plane, and the point at infinity on horizontal lines. Prove that there are real numbers a, b, e, h such that a 0 0, e 0 0, and the transformation maps (x, y, z) !

This factorization shows that s1 þ Á Á Á þ s v (8) is the degree d of g(x, 0) minus the degree of r(x). 3). We claim that the number of times, counting multiplicities, that y ¼ 0 intersects G ¼ 0 at infinity is n À d. We add this to the number of intersections in the Euclidean plane, which is d minus the degree of r(x) (by the previous paragraph). Then the total number of intersections in the projective plane is n minus the degree of r(x), as the theorem asserts. 54 I. Intersections of Curves To prove the claim, we count the intersections of y ¼ 0 and G ¼ 0 at infinity.

That degree is the least exponent on z in a nonzero term of G(x, 0, z). That exponent is n À d, since d is the largest exponent on x in a nonzero term of (6). We have established the claim in the previous paragraph. r We can now prove that any line intersects any curve of degree n that does not contain it at most n times in the projective plane, counting multiplicities. We need one preliminary observation. If a transformation mapping (x, y, z) to (x 0 , y 0 , z 0 ) takes homogeneous polynomials F(x, y, z) and G(x, y, z) to F 0 (x 0 , y 0 , z 0 ) and G 0 (x 0 , y 0 , z 0 ), then F is a factor of G if and only if F 0 is a factor of G 0 .

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