lemmaroutine

Generic complete grid with independent coordinates

Lemma·alg:generic-grid·§8.2 A complete grid with generic positions

Statement (paper)

For every fixed N,j≥1N,j\ge1, one can choose a complete NN-by-jj grid in the pruned graphs on a large set of indices so that its limiting vertices p1,…,pN∈V1p_1,\ldots,p_N\in V_1 and r1,…,rj∈V2r_1,\ldots,r_j\in V_2 have the following property. Select both coordinates of each vertex on a plane locus and its zz-coordinate on a curve locus. All the resulting Nd1+jd2Nd_1+jd_2 coordinates are algebraically independent over B0B_0. The other coordinates on a curve are algebraic over these coordinates and B0B_0. In particular, 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 selected coordinates and edge ratios belong to FF.

In plain English

For fixed N,j≥1N,j\ge1, the pruned graphs contain a complete NN-by-jj grid, on a large set of indices, whose limiting vertices p1,…,pN∈V1p_1,\ldots,p_N\in V_1 and r1,…,rj∈V2r_1,\ldots,r_j\in V_2 have all selected coordinates (Nd1+jd2Nd_1+jd_2 in total) algebraically independent over B0B_0. The remaining coordinates on curve loci are algebraic over these and B0B_0, so 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, and the selected coordinates and edge ratios lie in FF.

Why it's here

This is the genericity input for the transcendence arguments in Section 8: it provides a complete grid whose vertices behave like independent generic points of their loci. The proof counts complete grids (Jensen) and shows that each nonzero polynomial test kills only a power-saving fraction of tuples, so a diagonal choice avoids all tests at once. Later results use the grid to find a vertex of full transcendence degree d1d_1, embed function fields, and control grid ratios.
Classified routine:
A genericity lemma built from standard tools: Jensen-type counting of complete bipartite subgraphs, plus the fact that a nonzero polynomial vanishes on only a power-saving fraction of tuples (Schwartz–Zippel style), followed by a diagonal choice. It is technical input to the paper's transcendence arguments rather than a main new idea.

Uses

Nothing else in the paper

Used by

lemSome grid vertex is genericclaimSimple point on curve away from anchorclaimOther radicands avoid 0 and 4 at s_ellclaimGeneric points embed function fieldsclaimGrid ratios factor through constant defects

Inside the proof

claimPolynomials vanish rarely on lociclaimMany complete grids by JensenclaimPower saving for vanishing tests by inductionclaimDiagonal grid choice avoids all testsclaimCurve coordinates algebraic; transcendence conclusions

Sources

PaperOpen
Section
§8.2 A complete grid with generic positions
Page
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Label
alg:generic-grid
Line
sections/07-algebra.tex:197
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