claimroutineinferred_dep

Curve coordinates algebraic; transcendence conclusions

Claim·§8.2 A complete grid with generic positions·inside lemGeneric complete grid with independent coordinates

Statement (paper)

For a vertex lying on a curve locus given by an equation f(z,w)=0f(z,w)=0 with positive ww-degree, the coordinate ww is algebraic over B0(z)B_0(z) once zz is transcendental over B0B_0. Hence the other coordinates on a curve are algebraic over the selected coordinates and B0B_0, and the stated transcendence assertions hold, including trdeg⁡B0(r1,…,rj)B0(r1,…,rj,p1,…,pN)=Nd1\operatorname{trdeg}_{B_0(r_1,\ldots,r_j)} B_0(r_1,\ldots,r_j,p_1,\ldots,p_N)=Nd_1. The vertices and assigned edge ratios at each chosen finite grid lie in FnF_n, so their classes lie in FF.

In plain English

If a vertex lies on a curve locus f(z,w)=0f(z,w)=0 with positive ww-degree and zz is transcendental over B0B_0, then ww is algebraic over B0(z)B_0(z). So the non-selected coordinates on curves are algebraic over the selected coordinates and B0B_0, which gives the transcendence degree Nd1Nd_1 over B0(r1,…,rj)B_0(r_1,\ldots,r_j), and the chosen vertices and edge ratios lie in FF.

Why it's here

The independence of the selected coordinates from the diagonal grid choice makes each curve's zz transcendental, so the curve equation forces ww to be algebraic over it. Counting the Nd1Nd_1 independent coordinates of p1,…,pNp_1,\ldots,p_N over B0(r1,…,rj)B_0(r_1,\ldots,r_j) then gives the joint transcendence degree. The membership in FF holds because everything at each finite grid lies in FnF_n, which finishes the parent lemma.
Classified routine:
This is standard field-theory bookkeeping: a nontrivial polynomial relation f(z,w)=0 of positive w-degree makes w algebraic over B_0(z), and transcendence degree is additive in towers. It follows directly from the generic grid choice in the preceding claim (claim-alg-generic-grid-4).

Uses

claimDiagonal grid choice avoids all tests

Used by

Nothing in the paper

Sources

PaperOpen
Section
§8.2 A complete grid with generic positions
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sections/07-algebra.tex:279
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Friction

  • info
    inferred_dep
    outline: "Curve coordinates algebraic; transcendence conclusions" rests on claim-alg-generic-grid-4