Geometry And Topology

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By Casey J.

ISBN-10: 1418182842

ISBN-13: 9781418182847

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Extra info for A treatise on the analytical geometry of the point, line, circle, and conical sections (1885)

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2). b) If β ∈ R and α ∈ Fq ∪ {∞} satisfy the equation ϕ(β) = ψ(α), then α ∈ R. Then the ramification locus of the tower F satisfies V (F) ⊆ {P | P is a place of F0 with x0 (P ) ∈ R}; in particular , V (F) is finite and moreover deg P ≤ #R. 3) P ∈V (F ) Proof. Let P ∈ V (F). There is some n ≥ 0 and a place Q of Fn lying above P such that Q is ramified in the extension Fn+1 /Fn . Let P := Q ∩ Fq (xn ) denote the place of Fq (xn ) lying below Q, and consider the following diagram: 21 A. Garcia and H.

Considering the examples of asymptotically good towers in Section 4 and Section 5 one notes in all cases that degX f (X, Y ) = degY f (X, Y ). 3. (see [23]). Let F = (F0 , F1 , F2 , . ) be a recursive tower over Fq defined by the equation f (X, Y ) = 0. , the tower F is asymptotically bad. Proof. We set a := degY f and b := degX f . 8) that [Fn+1 : Fn ] = a and hence we have (for all n ≥ 1): [Fn : F0 ] = an and [Fn : Fq (xn )] = bn . 1) attached to the tower F is “skew”. Now we distinguish the cases a > b and a < b.

Stichtenoth Let α ∈ S and consider the solutions β ∈ F8 of the equation β 2 + β = α + 1 + α−1 . Then one has β 2 + β = α + 1 + α−1 , β 4 + β 2 = α2 + 1 + α−2 and β 8 + β 4 = α4 + 1 + α−4 . Adding these equations we obtain β 8 + β = α4 + α2 + α + 1 + α−1 + α−2 + α−4 = α−4 (α8 + α6 + α5 + α4 + α3 + α2 + 1) = α−4 (α6 + α5 + α4 + α3 + α2 + α + 1) = 0. 13 is satisfied. It is also clear that β = 0, 1, since otherwise α + 1 + α−1 = 0 and then α ∈ F8 . 13 holds. 13. Next we determine the ramification locus V (W4 ) of the tower W4 .