$p$-adic Integral Geometry

Antonio Lerario; Avinash Kulkarni

REVIEW 4 major objections 4 minor 20 references

$p$-adic Integral Geometry

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves a p-adic integral geometry formula: averaging the volume of the intersection of two p-adic projective algebraic sets over GL_{n+1}(Z_p) equals the product of their normalized volumes, exactly as in the real case.

desk verdict Novel p-adic integral geometry formula with real applications, but the submitted text has two fixable correctness gaps (a false volume bound and an unverified projective Hensel step) that make it a conditional accept. read the letter →

arxiv 1908.04775 v1 pith:E7U476JW submitted 2019-08-13 math.AG math.MGmath.NTmath.PR

Avinash Kulkarni , Antonio Lerario This is my paper · ORCID

classification math.AGmath.MGmath.NTmath.PR MSC 53C6511S80

keywords p-adicintegralgeometryformulavolumesprojectivealgebraicsetsrandompolynomialsHensel'slemmamodp^mpointcountsHaarmeasureonGL_n(Z_p)

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes a p-adic analogue of the classical integral geometry formula: for open compact subsets U and V of p-adic projective algebraic sets of dimensions a and b, averaging the volume of U ∩ gV over all g in GL_{n+1}(Z_p) equals the product of the individually normalized volumes of U and V. The identity mirrors the real projective formula exactly, with the same codimension arithmetic c = n − (n − a) − (n − b). If correct, the result turns a geometric density—the volume of an algebraic set—into a computable average of intersection counts with random linear subspaces, which the authors use to recover a classical bound on the number of points in the modulo p^m reduction of a projective set and to compute the expected number of solutions of random p-adic polynomial systems. The key step is a quantitative linear approximation: on most small p-adic balls, an algebraic intersection has the same point count as the intersection of the corresponding tangent spaces.

What carries the argument

The load-bearing device is the Linear Approximation Lemma (Lemma 33). It states that if X,Y ⊂ P^n are algebraic sets with local equations f_x, f_y at smooth points x,y, and if the normalized Jacobian determinant satisfies |J(f_x(x), f_y(y))|$_p^{2}$ > $p^{{-m}}$, then the number of points in X∩Y inside balls of radius $p^{{-m}}$ around x and y equals the number of points in the intersection of the tangent spaces T_xX and T_yY inside those balls. This is proved from Hensel's lemma and the p-adic implicit function theorem. The lemma lets the proof of Theorem 35 replace the algebraic sets by their tangent spaces on a fine-enough partition of U and V, so the integral over GL_{n+1}(Z_p) becomes a finite sum of integrals of indicator intersections of linear subspaces—evaluated exactly by Lemma 31, a purely linear statement about random linear subspaces of complementary codimension.

What would settle it

Evaluate Theorem 35 for A equal to the conic $x0^{2}$ + $x1^{2}$ + $x2^{2}$ = 0 in $P^{2}$ over Q_3 and B a random line (so c = 0): enumerate GL_3(Z_3) modulo 3^m, average the number of intersection points, and compare with vol_1(A)/vol_1($P^{1}$); a mismatch for any m would refute the formula. Equivalently, test Lemma 33 on a p-adic ball where the Jacobian condition fails and check whether the algebraic intersection count still equals the tangent-space count.

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Extended reading notes

Core claim

The central discovery is that the classical formula $$ \int_G \frac{\operatorname{vol}_k(A\cap gB)}{\operatorname{vol}_k(P^k)}\,dg = \frac{\operatorname{vol}_a(A)}{\operatorname{vol}_a(P^a)}\cdot \frac{\operatorname{vol}_b(B)}{\operatorname{vol}_b(P^b)}, \qquad k=n-(n-a)-(n-b), $$ with G = GL_{n+1}(Z_p), holds verbatim in the p-adic setting for algebraic subsets (and open compact subsets) of P^n. The volume vol_k(Y) is defined as the limit of the $p^{{m(n-k)}}$-normalized measure of the $p^{{-m}}$-neighborhood of Y, which agrees with the k-dimensional Hausdorff measure and with the normalized limit of mod p^m point counts. The proof passes through a linear approximation lemma that allows the replacement of Y by its tangent space near smooth points, reducing the integral geometry statement to a purely combinatorial identity for intersections of linear subspaces over Z_p. As applications, the paper recovers, with a shorter proof, the classical bound vol_k(Y) ≤ deg(Y) and the corresponding mod p^m counting bound for equidimensional projective sets, and computes expectations for two random p-adic polynomial models: the standard Veronese model has exactly one expected zero in P^n, and the Mahler-basis model has $p^{{⌊log_p d⌋}}$/(1+$p^{{-1}}$) expected zeros in Z_p (with an explicit extension to all of Q_p).

Load-bearing premise

The formula rests on a local approximation step that replaces each algebraic set by its tangent space on small balls, and this step must survive the normalization of the defining equations by powers of a point's norm when working in projective space; if that replacement is wrong at any scale, the averaging identity does not follow.

Editorial extensions

If this is right

The volume of any p-adic projective algebraic set can be computed as the expected number of intersections with a random linear subspace of complementary dimension, making volumes accessible to finite modulo-p^m computation.
A simplified proof of the bound vol_k(Y) ≤ deg(Y) for a k-dimensional p-adic projective set follows by bounding the intersection-number integrand by the degree; when the volume is strictly smaller, a mod p^m counting bound of the form N_m(Y) ≤ d p^{mk} vol_k(P^k) also follows.
For the random system of n homogeneous degree-d polynomials in n+1 variables with i.i.d. uniform Z_p coefficients, the expected number of projective zeros is exactly 1, and the zeros are uniformly distributed on P^n.
For the random Mahler-basis polynomial f(t) = Σ ξ_k binom(t,k), the expected number of zeros in Z_p is p^{⌊log_p d⌋}/(1+p^{-1}), recovering Evans' theorem, and the total expectation over Q_p is (p^{⌊log_p d⌋} + |d|_p p^{-1})/(1+p^{-1}).
Because the formula holds for open compact subsets and for smooth loci, it also computes volumes of the smooth part of singular sets, which equals the volume of the whole set.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The bound vol_k(Y) ≤ deg(Y) is one-sided; a natural conjecture, not addressed in the paper, is that equality occurs exactly for unions of linear subspaces, mirroring the extremal behavior of the real and complex Crofton inequalities.
The proof of the arc-length formula (Lemma 37) suggests a p-adic coarea formula for higher-dimensional smooth maps: if a p-adic analytic embedding has Jacobian of constant absolute value on a domain, its image volume is that constant times the domain volume. Such a formula would make Veronese-volume computations for systems of several equations routine.
The same linearization strategy should yield integral geometry formulas on other p-adic homogeneous spaces, such as Grassmannian intersections, because the only group-dependent ingredient is the linear-subspace averaging lemma.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know this paper is a genuine first: a p-adic analogue of the integral geometry formula for projective algebraic sets, averaging intersection volume under GL_{n+1}(Z_p). If it holds, it is a useful tool: it recovers Oesterlé's point-counting bound and Evans' expected zeros, and extends Evans to the whole p-adic line. The Veronese isometry (Prop. 38) is a neat result on its own, and the Mahler Veronese volume computation is explicit and checkable.

The paper deserves careful reading because the main theorem is new and the applications are real. The structure is sensible: prove a linear approximation lemma, then integrate over the group.

Now the soft spots, in order of seriousness.

First, Corollary 3 is simply false as stated. It claims vol_k(Y) ≤ d for any algebraic set Y of dimension k and degree d, but for Y = P^k the volume is (1-p^{-(k+1)})/(1-p^{-1}) > 1 = d. The proof says the integrand is bounded by degree for almost all g, which may be true, but the conclusion does not follow for P^k. This is a real error in the statement, though it may be an over-sharp bound they do not strictly need.

Second, Lemma 33 (Linear Approximation Lemma) is the load-bearing step for Theorems 34 and 35, and the proof as written is not complete. It invokes Hensel's lemma on a system of n homogeneous equations in n+1 variables, but Hensel's lemma in the form stated (Lemma 18) is for a square system on Z_p^n. The proof never chooses an affine chart, never dehomogenizes, and never checks that the normalized projective Jacobian condition |J|^2 > p^{-m} implies the affine condition needed for Hensel. This is not a routine omission: the normalization by ||x||^{δ} could easily fail to be an affine unit, and the common ball U_x = U_y might cross a coordinate hyperplane. The authors may be able to fix it by a chart-by-chart argument, but as written the central formula rests on an unverified assertion.

Third, in the proof of Theorem 35, they pick L ≃ P^c and apply Theorem 34, but Theorem 34 requires a linear subspace of dimension n - c. Dimension bookkeeping is off. Again, likely fixable, but it suggests the projective dimensional calculations need a careful pass.

There is no circularity—the formula is proved from standard tools, and no fitting appears. The citations to Serre, Oesterlé, Evans are appropriate.

Bottom line: this paper is for people working in p-adic geometry and random p-adic polynomial systems. The central result is new and plausible, but the submitted version has a false corollary and two proof gaps. It deserves a serious referee, but the referee should ask for a rewrite: fix Corollary 3, prove Lemma 33 with an explicit affine chart, and repair the dimension argument in Theorem 35. I would engage with it.

Referee Report

4 major / 4 minor

Summary. The paper develops a p-adic analogue of the classical integral geometry formula. For p-adic projective algebraic sets A,B of dimensions a,b in P^n, it claims that the average, over g in GL_{n+1}(Z_p), of the normalized volume of the intersection A∩gB equals the product of the normalized volumes of A and B. The proof strategy is to reduce the intersection count to the intersection of tangent spaces via a p-adic linear approximation lemma, to integrate the resulting linear intersections using a lemma on linear subspaces, and then to pass from point counts to volumes via a relation with modulo-p^m reduction. The paper also applies the formula to reprove a result of Oesterlé on congruence points, to compute expected numbers of zeros of random p-adic polynomial systems, and to recover and extend Evans' theorem for Mahler-basis random polynomials. The main novelty is the projective p-adic integral geometry formula itself and the applications to random p-adic equations.

Significance. If the central formula and its proof are completed, the paper would supply a genuine p-adic counterpart to the classical kinematic formula, with credible applications to random p-adic systems and to congruence-point estimates. The authors work with standard tools (Hensel's lemma, the p-adic implicit function theorem, Haar measure, and Oesterlé–Serre volume theory), and the main identity is parameter-free and falsifiable through its consequences, such as the expected number of zeros in the Veronese model. The paper is not circular: it proves the averaging formula and then applies it. However, the submitted text contains several load-bearing gaps and at least one demonstrably false statement, so the main theorem is not yet established as written.

major comments (4)

[§1.2, Corollary 3] The assertion vol_k(Y) ≤ d is false as stated. For Y = P^k linearly embedded in P^n, the degree is d = 1, while the paper's own computation gives vol_k(P^k) = (1 - p^{-(k+1)})/(1 - p^{-1}) = 1 + p^{-1} + ... + p^{-k} > 1. The displayed integral identity in the preceding paragraph yields the corrected bound vol_k(Y) ≤ d · vol_k(P^k), or equivalently vol_k(Y)/vol_k(P^k) ≤ d; the factor vol_k(P^k) is missing in the corollary and in the proof.
[§4, Lemma 33] The proof of Lemma 33 is not complete. Lemma 18 is an affine Hensel lemma for a square system f: Z_p^n → Z_p^n, but the system f_x ∪ f_y consists of n homogeneous polynomials in n+1 variables, so the displayed matrix J(f_x(x), f_y(y)) is n×(n+1) and 'det J' is not defined without choosing a chart and a maximal minor. The proof does not construct the affine chart, does not verify that the normalized projective Jacobian |J|_p agrees up to a p-adic unit with the affine Jacobian determinant required by Hensel's lemma, and does not show that the condition |J|^2 > p^{-m} implies the hypothesis ‖F(a)‖ < |J_F(a)|^2 of Lemma 18. Since Theorems 34 and 35 replace the algebraic sets by their tangent spaces precisely through this lemma, the p-adic integral geometry formula is currently not established.
[§4, Theorem 35] The proof of Theorem 35 says 'picking L ≃ P^c' and then applies Theorem 34. But Theorem 34, for a variety of dimension c in P^n, requires a linear subspace of dimension n - c, not dimension c. In addition, the statement of Theorem 34 itself appears to have a factor error: for U = P^a, L = P^{n-a}, the integral ∫ #(U∩gL)dg equals 1, while vol(φ^{-1}(P^a)) = 1 - p^{-(a+1)} but vol(P^a) = (1 - p^{-(a+1)})/(1 - p^{-1}). The final appeal to equation (3.2) suggests the intended identity is the normalized version vol_c(U) = vol(P^c)·∫#(U∩gL)dg, not the displayed vol(φ^{-1}(U)) = vol(P^a)·∫#(U∩gL)dg.
[§4, Theorem 34 proof, measure estimate] The bound µ(∪_{g∈Z}B(g,p^{-ℓ})) ≤ C p^{-ℓ dim(Z)} appears to confuse dimension with codimension. If Z is a proper algebraic subset of GL_N of dimension d, a tubular neighbourhood of radius p^{-ℓ} has measure of order p^{-ℓ(N-d)}, not p^{-ℓ d}; for a hypersurface with d = N-1 this is p^{-ℓ}, not p^{-ℓ(N-1)}. The proof needs an error term that vanishes as ℓ → ∞, and the estimate as written does not provide it. The same issue is compressed in Theorem 35 as µ(Z_ℓ) ≤ O(p^{-ℓ}).

minor comments (4)

[§2.1, Notation] The symbol R_m is used in the notation list before the definition of R_m as Z/p^mZ is made; define it at first use. Also, 'minumum' should be 'minimum'.
[§5.1, Corollary 39] The random polynomial is written with coefficients ξ_{1,α} but the index i runs from 1 to n; the coefficients should be ξ_{i,α}.
[§5.2, Theorem 47] There are typos in the statement: 'varuable' and 'unifomrly' should be 'variable' and 'uniformly'.
[§1.3, Theorem 5] The annulus in part (2) is written as '1/p^m Z_p\ 1/p^{m-1} Z_p'; adding parentheses, for example (1/p^m)Z_p \ (1/p^{m-1})Z_p, would remove the parsing ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the p-adic integral geometry formula is derived from Hensel/IFT, linear-subspace averaging, and volume-counting identities.

full rationale

The paper's central result, Theorem 35, is derived by combining the Linear Approximation Lemma 33 (based on Hensel's lemma and the p-adic implicit function theorem), the linear-subspace averaging Lemma 31, and the volume-counting identity of Corollary 25. None of these ingredients is the target formula, and no parameter is fitted to data and then renamed as a prediction. The citations to Serre, Oesterlé, Howard, Evans, and Weil are background or external benchmarks; they are not self-citations and are not used to assume the p-adic integral geometry formula. The only self-citation, BKL18, appears in the introduction as contextual motivation and is not load-bearing. The apparent dimensional slip in the proof of Theorem 35, where 'L ≃ P^c' should be 'L ≃ P^{n-c}', is a correctness/typo issue, not a circularity; the surrounding argument explicitly uses Lemma 31 with codimensions summing to n, which requires dim L = n-c. Any concern that Lemma 33's projective Hensel step is insufficiently justified is a rigor gap about chart-dependent Jacobians, not a reduction of the claimed result to its own inputs. Thus the derivation is self-contained in the relevant circularity sense.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no free parameters and no invented entities. It relies on standard results in p-adic analysis and on the prior existence of p-adic volumes as established by Serre and Oesterlé. The central derivation is self-contained given these background facts.

assumptions (5)

standard math Hensel's lemma in several variables
Used in Lemma 33 to lift a solution modulo p^m to a unique root in a ball, and in Proposition 20 to construct isometries between an algebraic set and its tangent space.
standard math Igusa's p-adic implicit function theorem
Used in Proposition 20 to obtain analytic coordinates; cited from [Igu00, Theorem 2.2.1].
domain assumption Existence and finiteness of the limit defining p-adic volume
The volume in Definition 22 and the identity in Corollary 25 assume the limit lim N_m(Y)/p^{mk} exists and is finite for algebraic sets; this is attributed to Serre [Ser81] and Oesterlé [Oes82].
standard math Haar measure on GL_{n+1}(Z_p) is normalized and invariant
The integral geometry formula integrates against this measure; standard property of compact p-adic groups.
standard math Stratification of algebraic sets into finitely many Q_p-analytic manifolds
Proved in Proposition 10 from the Jacobian criterion and used to define strata for the volume and the linear approximation arguments.

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Pith. "Pith review of $p$-adic Integral Geometry." pith.science (2026). https://pith.science/paper/E7U476JW

@misc{pith2026190804775,
  author       = {Pith},
  title        = {Pith review of: $p$-adic Integral Geometry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E7U476JW}},
  note         = {Machine review of arXiv:1908.04775}
}

abstract

We prove a $p$-adic version of the Integral Geometry Formula for averaging the intersection of two $p$-adic projective algebraic sets. We apply this result to give bounds on the number of points in the modulo $p^m$ reduction of a projective set (reproving a result by Oesterl\'e) and to the study of random $p$-adic polynomial systems of equations.

Figures

Figures reproduced from arXiv: 1908.04775 by the authors.

**Figure 1.** A depiction of the 3-adic unit circle S 1 in Z 2 3 . Each cell represents an open ball of radius 1 3 centered at the indicated point. The union of all the unshaded cells is S 1 . The unit −1 ∈ Z × 3 acts by reflecting the diagram through the origin. Note that S 1 has non-zero measure inside Z 2 3 , as well as the pullback ϕ −1 (U) of any non-empty open subset of P 1 . Third, all this discussion on volumes and the p-… view at source ↗

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Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Probabilistic analysis of the G rassmann condition number

Dennis Amelunxen and Peter B\" u rgisser. Probabilistic analysis of the G rassmann condition number. Found. Comput. Math. , 15(1):3--51, 2015

work page 2015

[2] [2]

Hard L efschetz theorem for valuations, complex integral geometry, and unitarily invariant valuations

Semyon Alesker. Hard L efschetz theorem for valuations, complex integral geometry, and unitarily invariant valuations. J. Differential Geom. , 63(1):63--95, 2003

2003

[3] [3]

Condition , volume 349 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]

Peter B\" u rgisser and Felipe Cucker. Condition , volume 349 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer, Heidelberg, 2013. The geometry of numerical algorithms

work page 2013

[4] [4]

Andreas Bernig and Joseph H. G. Fu. Hermitian integral geometry. Ann. of Math. (2) , 173(2):907--945, 2011

2011

[5] [5]

On the G eometry of the S et of S ymmetric M atrices with R epeated E igenvalues

Paul Breiding, Khazhgali Kozhasov, and Antonio Lerario. On the G eometry of the S et of S ymmetric M atrices with R epeated E igenvalues. Arnold Math. J. , 4(3-4):423--443, 2018

work page 2018

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On the distribution of points in projective space of bounded height

Kwok-Kwong Choi. On the distribution of points in projective space of bounded height. Trans. Amer. Math. Soc. , 352(3):1071--1111, 2000

work page 2000

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A multivariable hensel's lemma

Keith Conrad. A multivariable hensel's lemma. Available at https://kconrad.math.uconn.edu/blurbs/gradnumthy/multivarhensel.pdf

work page

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How many zeros of a random polynomial are real? Bull

Alan Edelman and Eric Kostlan. How many zeros of a random polynomial are real? Bull. Amer. Math. Soc. (N.S.) , 32(1):1--37, 1995

work page 1995