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arxiv: 2511.12812 · v2 · submitted 2025-11-16 · 🧮 math.AG

Rational Points in Weighted Projective Spaces over Finite Fields

Sajad Salami , Tanush Shaska This is my paper

Pith reviewed 2026-05-17 21:32 UTC · model grok-4.3

classification 🧮 math.AG
keywords weighted projective spacesrational pointsfinite fieldsBurnside's lemmazeta functionssmooth and singular locigcd computations
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The pith

Three notions of rational points on weighted projective spaces coincide over finite fields.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that three different definitions of F_q-rational points on a weighted projective space actually describe exactly the same set of points. If this holds, then researchers can pick whichever definition is easiest for a given calculation and still obtain the correct count. The work supplies explicit formulas for these counts that depend only on the weights, the field size, and gcd calculations, obtained by applying Burnside's lemma to the natural group action. It further separates the points into smooth and singular parts and proves that the zeta function of the space is a rational function that factors according to this separation.

Core claim

We establish the equivalence of three notions of F_q-rational points on weighted projective spaces P_w^n and derive explicit combinatorial formulas for their enumeration, leveraging Burnside's lemma and gcd computations. We further derive formulas for point counts under weight normalization, providing closed expressions for the singular and smooth loci. Our results confirm the rationality of the zeta function Z(P_w^n, t) via a finite product formula and reveal a canonical multiplicative decomposition aligning with the stratification into smooth and singular loci.

What carries the argument

Burnside's lemma applied to the multiplicative group action on the weighted affine cone, with orbit sizes computed via gcds of the weights.

If this is right

  • Point counts on these spaces reduce to combinatorial calculations involving only the weights and the field order.
  • The zeta function factors as a finite product that respects the decomposition into smooth and singular strata.
  • Normalized weights yield separate closed formulas for the number of smooth points and the number of singular points.
  • The multiplicative decomposition of the zeta function holds uniformly for any choice of weights.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The orbit-counting method may extend directly to other toric varieties whose singularities are controlled by similar weight data.
  • The explicit formulas make it feasible to tabulate point counts for many weights and fields without resolving the equations geometrically.
  • The stratification-based factorization suggests that other arithmetic invariants, such as the number of curves through the points, might also decompose along smooth and singular loci.

Load-bearing premise

The three notions of rational points coincide for arbitrary positive integer weights and any finite field.

What would settle it

Compute the three candidate point counts directly for a small weight vector such as (1,2,3) and a small field such as F_3; if any two counts differ, the claimed equivalence fails.

read the original abstract

We establish the equivalence of three notions of $\mathbb{F}_q$-rational points on weighted projective spaces $\mathbb{P}_{\mathbf{w}}^n$ and derive explicit combinatorial formulas for their enumeration, leveraging Burnside's lemma and gcd (greatest common divisor) computations. We further derive formulas for point counts under weight normalization, providing closed expressions for the singular and smooth loci. Our results confirm the rationality of the zeta function $Z(\mathbb{P}_{\mathbf{w}}^n, t)$ via a finite product formula and reveal a canonical multiplicative decomposition aligning with the stratification into smooth and singular loci. These contributions advance the arithmetic theory of weighted projective spaces over finite fields, with computational examples illustrating the formulas for specific weights and fields.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes the equivalence of three notions of F_q-rational points on weighted projective spaces P_w^n for arbitrary positive integer weights. It derives explicit combinatorial formulas for their enumeration by applying Burnside's lemma to the weighted G_m-action on F_q^{n+1} minus zero, reducing fixed-point conditions to gcd computations. The paper further supplies formulas after weight normalization, closed expressions separating singular and smooth loci, and proves that the zeta function Z(P_w^n, t) is rational via a finite product with a multiplicative decomposition aligned to the smooth/singular stratification. Computational examples for specific weights and fields are included to illustrate the results.

Significance. If the derivations hold, the work advances the arithmetic theory of weighted projective spaces over finite fields by supplying explicit, computable point-count formulas that are polynomials in q. The confirmation of zeta-function rationality and the canonical multiplicative decomposition over strata are useful structural results. The unconditional applicability of Burnside's lemma to the finite-group action, together with the direct reduction to gcds, yields concrete combinatorial expressions without additional hypotheses, which is a clear strength for both theory and computation.

minor comments (3)
  1. [Abstract] Abstract: the claim of 'closed expressions for the singular and smooth loci' would be clearer if the abstract briefly indicated that these are finite products or polynomials in q.
  2. [Zeta function section] The section deriving the zeta-function formula: explicitly write the finite product in terms of the stratum point counts so that rationality follows immediately from the polynomial nature of the counts.
  3. [Examples section] Computational examples: include at least one small case where the formula is compared directly to exhaustive enumeration of orbits to allow immediate verification.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript, the favorable assessment of its significance, and the recommendation for minor revision. The work is correctly characterized as establishing equivalences of rational-point notions on weighted projective spaces, deriving explicit combinatorial counts via Burnside's lemma and gcd reductions, and confirming zeta-function rationality through a finite product formula with a smooth/singular decomposition.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via standard Burnside and gcd

full rationale

The paper applies Burnside's lemma directly to the finite group action of F_q^* on F_q^{n+1} minus zero under the weighted G_m scaling, with fixed-point conditions for each group element reducing to lambda^{w_i}=1 for nonzero coordinates, which is resolved by explicit gcd(w_i, ord(lambda)) counts. These steps invoke only unconditional group theory and arithmetic, independent of the target point-count formulas or zeta-function rationality. Equivalence of the three notions of F_q-rational points follows from the resulting explicit combinatorial expressions without any self-definition, fitted-parameter renaming, or load-bearing self-citation. The stratification into smooth/singular loci and finite-product zeta formula are direct consequences of the same counts, with no reduction to prior author results or ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard mathematical tools rather than new postulates. Burnside's lemma is invoked for enumeration and gcd computations for weight interactions; no free parameters or invented entities are indicated.

axioms (2)
  • standard math Burnside's lemma applies to the natural action of the multiplicative group on the weighted projective space over F_q
    Used to obtain the explicit combinatorial count of rational points.
  • domain assumption The three notions of F_q-rational points are well-defined for any positive integer weight vector w
    Required for the claimed equivalence to make sense.

pith-pipeline@v0.9.0 · 5411 in / 1297 out tokens · 32719 ms · 2026-05-17T21:32:06.200014+00:00 · methodology

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Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

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