Pith Number

pith:C3RHVY35

pith:2026:C3RHVY35BCUT3KT2KX3ID4R437

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Notes on Lie derivatives, algebraic D-varieties, and Ax's theorem

Anand Pillay

Lie derivatives on algebraic D-varieties correspond to linear differential equations on cotangent spaces at sharp points.

arxiv:2605.12670 v1 · 2026-05-12 · math.AG · math.LO

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Claims

C1strongest claim

We discuss the relationship between Lie derivatives and the linear differential equations on cotangent spaces of algebraic D-varieties at sharp points.

C2weakest assumption

The definitions and properties of algebraic D-varieties and sharp points are taken as given from prior literature without re-derivation in the notes.

C3one line summary

The notes connect Lie derivatives to differential equations on algebraic D-varieties and present an account of Ax's theorem as an entry point for students.

References

11 extracted · 11 resolved · 0 Pith anchors

[1] Ax, On Schanuel’s conjectures, Annals of Math 1971

[2] Buium, Differential algebraic groups of finite dimension, Springer 1992 1992

[3] Kirby, The theory of the exponential differential equations of semia- belian varieties, Selecta Math, 15 (3009), 445 486

[4] Kolchin, Abelian extensions of differential fields, American J 1960

[5] Johnson, Differential dimension polynomials and a fundamental the- orem on differential modules, Am 1969

Formal links

1 machine-checked theorem link

Receipt and verification

First computed	2026-05-18T03:09:50.234127Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

16e27ae37d08a93daa7a55f681f23cdfc017c457e725b626f8fa41c6af9e4675

Aliases

arxiv: 2605.12670 · arxiv_version: 2605.12670v1 · doi: 10.48550/arxiv.2605.12670 · pith_short_12: C3RHVY35BCUT · pith_short_16: C3RHVY35BCUT3KT2 · pith_short_8: C3RHVY35

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/C3RHVY35BCUT3KT2KX3ID4R437 \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 16e27ae37d08a93daa7a55f681f23cdfc017c457e725b626f8fa41c6af9e4675

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "b25a8b8428b7769005761265c024f663a930d9a9a0c54614b13541255a0c59e8",
    "cross_cats_sorted": [
      "math.LO"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.AG",
    "submitted_at": "2026-05-12T19:19:38Z",
    "title_canon_sha256": "3e58b65dd55b12f87069e439068c4d53862e7c057b26b2d0f1e12ab6ff694bb2"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.12670",
    "kind": "arxiv",
    "version": 1
  }
}