bibkey: bishnoi2017finitegrid authors: Anurag Bishnoi, Pete L. Clark, Aditya Potukuchi and John R. Schmitt year: 2017 title: On zeros of a polynomial in a finite grid doi: 10.48550/arXiv.1508.06020 url: https://arxiv.org/abs/1508.06020v2 claim: Corollary 6.8 gives the minimum n(q-1)+1 for affine hyperplane blocking sets in AG(n,q); its n=2 case is the all-line minimum 2q-1 over every finite field. strata_touched:
- D5/S3/Geometry/FiniteGeometry/AffineBlockingBound license: citation-only triage: anchor
The affine-plane blocking minimum
Anurag Bishnoi, Pete L. Clark, Aditya Potukuchi and John R. Schmitt, On zeros of a polynomial in a finite grid, arXiv:1508.06020v2, 12 June 2017. Primary text: https://arxiv.org/pdf/1508.06020v2.
Corollary 6.8, PDF page 16, states that a smallest blocking set in AG(n,q) has n(q-1)+1 points. Section 6.2 defines blocking as meeting every affine hyperplane over the finite field of order q. There is no prime-order or odd-characteristic restriction. The paper attributes the result to Jamison and to Brouwer and Schrijver. In dimension two, affine hyperplanes are lines, and the minimum is 2q-1.
In the formal statement the plane is the native product F x F, and its lines are the fibres phi(p)=c of all nonzero F-linear maps phi to F, with every offset c in F. A nonzero linear functional on this plane has a nonzero coefficient and is surjective; each fibre is a translate of its one-dimensional kernel. Thus these fibres describe the affine lines in the source’s n=2 setting, including the zero offset.
The proof chooses b0 in a blocking set B. For b in B other than b0, form the factor 1-(b.1-b0.1)u-(b.2-b0.2)v, and multiply these factors to obtain P. At the zero covector P is one. For a nonzero covector (u,v), the native linear map phi(x,y)=ux+vy is nonzero. Blocking at phi(b0)+1 gives a point b distinct from b0 with phi(b-b0)=1, so a factor vanishes. Consequently P vanishes at every other covector. Its total degree is at most |B|-1. Mathlib’s finite-field evaluation-sum theorem excludes this singleton support at degree less than 2(q-1), proving |B| at least 2q-1.
The two coordinate axes meet every fibre of every nonzero functional: division by a nonzero coefficient gives a point with any prescribed image. The axes each have q points and share only the origin, attaining 2q-1.
The formal theorem covers the n=2 case only. It does not formalize the full Alon–Furedi theorem or Corollary 6.8 in arbitrary dimension, nor assert a result for arbitrary incidence planes, nonfield rings, selected direction families or projective blocking sets. The singleton-support argument uses Mathlib’s evaluation-sum theorem; the finite-geometry result remains literature-attested, without an originality claim.
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DOI resolver: https://doi.org/10.48550/arXiv.1508.06020 Declared URL: https://arxiv.org/abs/1508.06020v2 Corollary 6.8, PDF page 16; blocking-set definition in Section 6.2, PDF pages 15-16.