Smith normal form of affine point–line incidence over prime-power rings
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What are the integer Smith factors, including their multiplicities, of affine point–line incidence matrices over Z/pᵏZ? The recorded work treats depths k=2 and k=3 for every prime p.
Why it matters and what counts as progress
Why it matters: Smith factors retain integral incidence information that finite-field ranks and partial depth profiles do not determine.
What counts as progress: Determine the complete profile at a new depth or prove uniform multiplicity formulas, including exceptional small primes.
How the problem developed
- 2006 · Field case [1]
Peter Chandler, Peter Sin and Qing Xiang Determined invariant factors for point–subspace incidence matrices in finite projective and affine spaces over finite fields; this is the field setting, distinct from incidence over prime-power rings.
- October 2026 · Depth two result [2]
Oleksiy Babanskyy An author-hosted manuscript reports the complete p-primary Smith profile for the depth-two affine Hjelmslev–Radon incidence cokernel, with transfer, nonsplitting and carry-geometry results.
- October 2026 · Depth three result [3]
Oleksiy Babanskyy A separate manuscript reports depth-three p-primary Smith profiles using Newton-minor geometry for affine Hjelmslev–Radon incidence.
AI contributions 2
Depth-Two Smith Profiles and Carry Geometry for Affine Hjelmslev–Radon Incidence
Babanskyy determines the complete p-primary Smith profile of the integer affine point–line incidence matrix over Z/p²Z for every prime p, including the nonunit factors p, p² and p³ and their multiplicities. The argument also gives a nonsplit transfer extension and a two-chart carry presentation.
Depth-Three Smith Profiles and Newton-Minor Geometry for Affine Hjelmslev–Radon Incidence
Babanskyy reports every nonzero Smith factor of the integer affine point–line incidence matrix over Z/p³Z for every prime p. The profile reaches exponent p⁵ and has six multiplicities, with polynomial formulas for odd p and explicit p=2,3 rows. The new step resolves a q-Pascal common image through q-Newton minors and a cross-chart carry calculation.
Sources
- Chandler, Sin and Xiang, The Invariant Factors of the Incidence Matrices of Points and Subspaces in PG(n,q) and AG(n,q)
- Affine incidence Smith-profile manuscript
- Affine incidence Smith-profile manuscript
- AI contribution and author checking disclosure
- Project theorem and verification notes
- AI contribution and author checking disclosure
- Project theorem and verification notes
Website timeline
Publication dates and editorial changes.
AI-assisted work reports the depth-two affine incidence Smith profile
AI-assisted work reports the depth-three affine incidence Smith profile