By Professor Wolfram Decker, Professor Gerhard Pfister

A primary direction in Computational Algebraic Geometry is designed for younger scholars with a few historical past in algebra who desire to practice their first experiments in computational geometry. Originating from a direction taught on the African Institute for Mathematical Sciences, the e-book offers a compact presentation of the elemental idea, with specific emphasis on specific computational examples utilizing the freely on hand laptop algebra approach, Singular. Readers will speedy achieve the arrogance to start appearing their very own experiments.

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**Example text**

Xn ] is a Gr¨ obner basis of Ik with respect to >lp on K[xk+1 , . . , xn ], for k = 0, . . , n − 1. 2 for details. 59 Let I ⊂ K[x1 , . . , xn ] be an ideal, let A = V(I) be its vanishing locus in An (K), let 0 ≤ k ≤ n − 1, and let πk : An (K) → An−k (K), (x1 , . . , xn ) → (xk+1 , . . , xn ), be projection onto the last n − k components. Then πk (A) = V(Ik ) ⊂ An−k (K). 5 on Buchberger’s algorithm and field extensions, the ideal generated by Ik in the polynomial ring K[xk+1 , . . , , xn ] is the first elimination ideal of the ideal generated by I in K[x1 , .

In particular, V(φ(I)) and, thus, V(I) are nonempty. 71 Let 0 = I K[x1 , . . , xn ] be an ideal. 69 at each stage, we may suppose after a lower triangular coordinate change 1 x1 .. . → xn ∗ 0 .. 1 x1 .. . xn that the coordinates are chosen such that each nonzero elimination ideal Ik−1 = I ∩ K[xk , . . , xn ], k = 1, . . , n, contains a monic 44 The Geometry–Algebra Dictionary polynomial of type (k) (k) fk = xdkk + c1 (xk+1 , . . , xn )xdkk −1 + . . + cdk (xk+1 , .

Hence, K[A] is naturally a K–algebra. Next, observe that each morphism ϕ : A → B of algebraic sets gives rise to a homomorphism ϕ∗ : K[B] → K[A], g → g ◦ ϕ, of K–algebras. Conversely, given any homomorphism φ : K[B] → K[A] of K–algebras, one can show that there is a unique polynomial map ϕ : A → B such that φ = ϕ∗ . Furthermore, defining the notion of an isomorphism as usual by requiring that there exists an inverse morphism, it turns out that ϕ : A → B is an isomorphism of algebraic sets iff ϕ∗ is an isomorphism of K–algebras.