Improved Frobenius lifts for nondegenerate curves

Definitions and mathematical statement

Let $p$ be the characteristic prime of the intended finite-field curve setting. A Frobenius lift in a $p$-adic computation is an endomorphism, with the appropriate coefficient-Frobenius action, of a chosen lifted/completed coordinate algebra whose reduction modulo $p$ is $a\mapsto a^p$. The source does not specify the coefficient ring, completion, curve equation, or which coordinate is denoted $x$.

The proposed optimization is to delete additional points of a nondegenerate curve and prescribe the coordinate image \[ \varphi(x)=x^p \] when constructing its Frobenius lift. If $C$ is the intended curve and $D$ the extra removed locus, the construction would operate on $U=C\setminus D$ and its corresponding lifted algebra. The source gives neither $D$ nor a deletion rule. It also does not define nondegeneracy, for example by a Newton-polygon/toric condition, so no such condition is inferred.

The task is to test and analyze whether this deletion-plus-prescribed-coordinate construction improves the relevant Frobenius-lift method. A precise evaluation needs the baseline lift, admissible curves, permissible deleted points, exact meaning of a lift on the open curve, required precision, and performance criterion. Possible criteria include running time, sparsity, coefficient growth, pole orders, or precision loss, but none is prescribed. Nor does the source state how any curve invariant being computed is corrected for the deleted points. These are missing inputs to an algorithmic research program, not a fully specified universal improvement theorem.

Short English statement

Investigate whether allowing extra punctures and requiring $x\mapsto x^p$ makes Frobenius lifts for nondegenerate curves more efficient. The source supplies no concrete algorithm, input family, or improvement metric for an executable test.

The supplied summary mentions Kedlaya's method, toric/nondegenerate extensions, Castryck, Chtcherbakov, and sparse cohomology progress, but says no verified result settles this exact combined optimization. Those are reported partial-progress themes, not a theorem or benchmark. The 2006 source's current status needs primary-literature review.

Sources

• Local UnsolvedMath registry, permanent ID 8800014, source code catalogue entry, "Improved Frobenius lifts for nondegenerate curves," proposed by Kiran Kedlaya. Dataset curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0.
• IPAM, Workshop on Number Theory and Cryptography --- Open Problems (2006), Attachment 6220, PDF pages 30--32, logical slide 16, schedule locator https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule; pdf-and-legacy-ppt visual-audit extraction, accessed 2026-07-29.
• The supplied summary mentions Castryck, Chtcherbakov, and toric-decomposition literature without complete primary references or an exact optimization theorem.
• Source provenance reports 19 surviving attachments inspected, comprising 17 PDFs and two legacy PPT decks, excluding repetitions and solved definitions; this is supplied provenance, not a new inspection claimed here.
• Current primary-literature status audit and redistribution terms NEEDS_REVIEW for the publicly accessible author/publisher copy, separately from the dataset licence.