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Local-global principles for visibility of lattice points on parameterized curves

This paper establishes a local-global principle for the visibility of lattice points on parameterized curves, proving that global visibility is determined by local $p$ -adic visibility for weighted homogeneous families while demonstrating that this principle holds almost everywhere for polynomial families of the form $y=qP(x)$ , with specific quantitative bounds and extensions to visibility between arbitrary lattice points.

Original authors: Sneha Chaubey, Anwesh Ray

Published 2026-09-14

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Sneha Chaubey, Anwesh Ray

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a vast, infinite grid of dots stretching out in every direction, like the intersections of a city street map that never ends. In mathematics, these are called lattice points. A classic puzzle asks: if you stand at the very center of this grid, which of the other dots can you see directly? You can see a dot if a straight line drawn from your feet to that dot does not pass through any other dot first. If another dot blocks the view, the distant one is hidden. This simple rule of visibility turns out to be a powerful way to understand the hidden structure of numbers. For decades, mathematicians have known that whether a dot is visible depends entirely on the numbers that describe its position. If those numbers share a common factor, like both being even, the dot is hidden. If they share no common factor, the dot is visible. This is a local rule: you can check for visibility by looking at the numbers one by one, prime by prime, much like checking if a lock has a key for every single pin inside it.

The question that drives this new research is whether this local check is enough to guarantee a global truth. If a dot looks visible when you check it against every single prime number individually, does that mean it is truly visible from the center? For some very specific, highly symmetric families of curves, the answer is a definitive yes. But for others, the answer is more complicated. The researchers, Sneha Chaubey and Anwesh Ray, set out to map exactly where this local check works and where it fails. They studied families of curves that can be described by simple formulas, asking when a point on such a curve is visible from the origin. They found that for curves with a specific kind of scaling symmetry, the local check is perfect: if a point passes the test for every prime, it is globally visible. However, for curves defined by more complex polynomial formulas, this perfect match breaks down. There are points that pass every local test but are still hidden from view.

The team discovered that while these "false positives" exist, they are incredibly rare. In the grand scheme of all the points on these curves, the number of points that trick the local test is so small that they effectively vanish when you look at the big picture. The researchers proved that for a broad class of polynomial curves, the set of points where the local check fails has a density of zero. This means that if you were to pick a random point on such a curve, the chance that it is one of these deceptive, locally-visible-but-globally-hidden points is zero. The researchers also provided precise estimates for how many of these deceptive points exist as the grid gets larger, showing that they grow much slower than the total number of points.

To understand these results, one must first grasp the geometry of the curves they studied. Some of these curves are like straight lines or simple parabolas that stretch out from the origin. Others are more intricate, winding through the grid in ways that depend on the specific numbers in their formulas. The researchers introduced a concept of "weighted" visibility, where different directions on the grid are stretched or compressed. In these weighted worlds, the rules of visibility change, but the local-global principle often still holds if the stretching follows a consistent pattern. They showed that when the stretching is uniform and follows a specific mathematical rhythm, the local checks perfectly predict the global reality.

However, the story changes when the curves are defined by polynomials that are not simple powers. For example, a curve defined by a formula like $y = x^2 + x$ behaves differently than one defined by $y = x^2$ . The researchers proved that for the non-simple curves, the local-global principle fails. They constructed specific examples of points that are invisible from the origin but appear visible when checked against any single prime number. These points are the "defect" in the system. Yet, the researchers were able to show that these defects are sparse. They calculated that the number of such deceptive points grows at a rate that is significantly slower than the total number of points on the curve. In fact, for certain types of curves, the number of deceptive points grows so slowly that their proportion in the total population shrinks to nothing as the grid expands.

The study also explored what happens when the points on the curve do not fill the entire grid but are confined to a lower-dimensional shape, like a surface floating inside a higher-dimensional space. In these "sparse" families, the density of visible points can behave in surprising ways. Sometimes the density is a simple fraction, but other times it involves fractional powers that do not correspond to the dimension of the space in an obvious way. For instance, on a specific type of curved surface, the density of visible points is determined by a number that is not a whole integer, reflecting the complex way the points are distributed. This finding challenges the intuition that the density of visible points should always be a simple fraction related to the number of dimensions.

Finally, the researchers extended their work to a more general scenario: visibility between two arbitrary points on the grid, not just from the center. They showed that the same principles apply. If you stand at one lattice point and look at another, the question of whether you can see it depends on the difference between their positions. They proved that for the same symmetric families of curves, the local check works perfectly here too. If the displacement between two points passes the local test for every prime, then the second point is visible from the first. This generalization confirms that the local-global principle is a robust feature of these mathematical structures, holding true even when the perspective shifts away from the origin.

The work provides a complete picture of when local checks are sufficient and when they are not. It confirms that for a wide range of mathematical objects, the local behavior dictates the global outcome. But it also highlights the subtle exceptions where the local view can be misleading. By quantifying exactly how rare these exceptions are, the researchers have turned a potential flaw in the local-global principle into a precise mathematical fact. They have shown that while the principle can fail, the failure is so limited that it does not disrupt the overall order of the system. The results offer a deeper understanding of the distribution of lattice points and the intricate relationship between local arithmetic properties and global geometric visibility.

Technical Summary: Local-Global Principles for Visibility of Lattice Points on Parameterized Curves

1. Problem Statement and Motivation
The paper investigates the local-global principle for the visibility of lattice points on families of parameterized curves. In classical number theory, a point $P \in \mathbb{Z}^n$ is visible from the origin if the line segment connecting them contains no other lattice points, which is equivalent to $\gcd(P_1, \dots, P_n) = 1$ . This condition is local: $P$ is visible if and only if for every prime $p$ , not all coordinates are divisible by $p$ .

The authors generalize this to families of curves $\mathcal{Y} = \{(U_\alpha, \phi_\alpha)\}_{\alpha \in I}$ , where $\phi_\alpha: U_\alpha \to X \subseteq \mathbb{R}^n$ are rational parametrizations passing through the origin. A point on a branch is globally visible if it is the first integral point on that specific curve. It is $p$ -adically visible if no earlier integral point on the same curve has a strictly smaller $p$ -adic minimum valuation. The central question is whether global visibility is equivalent to $p$ -adic visibility for all primes $p$ (local detectability). Specifically, the authors ask:

When does the equality $B_{\text{vis}}(\mathcal{Y}) = \bigcap_p B_{\text{vis}, p}(\mathcal{Y})$ hold?
When the equality fails, what is the density of the "defect set" $B_{\text{bad}}(\mathcal{Y})$ (points that are locally visible everywhere but globally invisible)?
How do the global and local densities of visible points relate, particularly in sparse families where the branch locus does not fill the entire lattice?

2. Methodology
The paper employs a combination of algebraic geometry, analytic number theory, and combinatorial counting arguments:

Weighted Homogeneity: The authors utilize the action of the multiplicative group $\mathbb{G}_m$ on $\mathbb{R}^n$ with positive weights $w_i$ . Families satisfying $\phi_\alpha(dt) = d \cdot \phi_\alpha(t)$ are termed "positive-weight homogeneous."
Davenport's Lattice Point Estimate: For sparse families (where the branch locus lies on a lower-dimensional subvariety), the authors use Davenport's lemma to estimate the number of lattice points in regions defined by height bounds, relating the volume of the region to the count of integer points.
Möbius Inversion and Euler Products: To compute densities, the authors use Möbius inversion to isolate primitive points and analyze the convergence of Euler products formed by local densities.
Bombieri–Pila Estimates: For polynomial families that are not monomials, the authors employ the Bombieri–Pila theorem on the number of integral points on absolutely irreducible curves to bound the size of the invisible locus.
Schur's Theorem and Hensel Lifting: To construct counterexamples for non-monomial polynomials, the authors use Schur's theorem on prime divisors of polynomial values and Hensel's lemma to construct points that are locally visible but globally invisible.

3. Key Contributions and Results

A. Weighted Homogeneous Families

Local-Global Principle: Theorem 1.1 establishes that for families with positive-weight homogeneous parametrizations, visibility is locally detectable. A point is globally visible if and only if it is $p$ -adically visible for every prime $p$ .
Density Formulas: When the branch locus fills the positive integer lattice $\mathbb{Z}^n_{\ge 1}$ and the weights $w_i$ are coprime, the global density of visible points is $1/\zeta(W)$ , where $W = \sum w_i$ . This global density is exactly the product of the local densities $d_{\text{vis}, p} = 1 - p^{-W}$ (Theorem 1.3).
Geometric Interpretation: The authors connect weighted primitivity to the arithmetic of weighted projective stacks, showing that the weighted GCD corresponds to the order of the stabilizer in the stack.

B. Sparse Families
The paper analyzes families where the branch locus is a proper algebraic subset of the ambient space.

Sparse Radial Graph Families: For families defined by graphs of homogeneous polynomials (e.g., $(x, y, x^d y)$ ), the authors derive conditions under which the local-global principle holds and the density is $1/\zeta(r)$ , where $r$ is the dimension of the parameter space (Theorem 4.3).
Failure of Uniformity: Example 4.4 demonstrates that if the geometric conditions on the region (specifically condition 4.2 regarding the growth of the region's volume vs. its projections) are not met, the density exponent can change (e.g., from $1/\zeta(2)$ to $1/\zeta(d+1)$ ).
Rational Rays on Cones: For families defined by rational rays on affine cones (e.g., Pythagorean triples $X^2+Y^2=Z^2$ $X^{2} + Y^{2} = Z^{2}$ ), the global density depends on the growth rate of primitive points on the variety.
- For the Pythagorean cone, primitive points grow linearly ( $X$ ), while total points grow as $X \log X$ , resulting in a global density of 0. The local density is $1 - 1/p$ , and the Euler product $\prod (1-1/p)$ vanishes, consistent with the global density (Proposition 4.9).
- For the quadratic cone $XZ=Y^2$ , the density is also 0.
- For rank-one $2 \times 2$ matrices, the density is $1/\zeta(2)$ , despite the variety having dimension 3.
- For the quadratic Veronese cone in 3 variables, the density is $1/\zeta(3/2)$ . The authors emphasize that the exponent in the zeta function is determined by the growth of the primitive counting function relative to the height, not necessarily the dimension of the variety, and can be fractional.

C. Polynomial Families (Non-Homogeneous)

Monomial vs. Non-Monomial: For families of the form $y = qP(x)$ , the local-global principle holds if and only if $P(x)$ is a monomial (Theorem 1.5).
Defect Set Density: If $P(x)$ is not a monomial, the local-global principle fails. However, the set of points where it fails (the defect set) has density zero relative to the branch locus (Theorem 5.3).
Quantitative Bounds:
- The authors improve the known upper bound for the number of invisible points $U(P; X)$ for separable polynomials of degree $d$ from $O(X^{2 - \frac{1}{2d-1} + \epsilon})$ to $O(X^{\frac{3}{2} + \frac{1}{2d} + \epsilon})$ (Theorem 1.6).
- They provide lower and upper bounds for the size of the defect set $D_P(X)$ . For non-monomials, $D_P(X) \gg X$ . For specific forms like $P(T) = T^r(m+nT)$ , they show $D_P(X) \asymp X \log X$ and provide upper bounds of $O(X^{7/4+\epsilon})$ (or $O(X^{5/3+\epsilon})$ for the cubic case $r=2$ ) (Theorem 1.7).

D. Mutual Visibility
The theory is extended to visibility between two arbitrary lattice points $A$ and $B$ . By translating the problem to the displacement vector $B-A$ , the authors show that the local-global principle and density results for weighted homogeneous families extend to "signed" families covering the whole lattice (Theorem 7.2). The density of visible pairs is $1/\zeta(W)$ .

4. Significance and Claims
The paper claims to develop a comprehensive local-global theory for lattice point visibility that unifies classical results with new phenomena arising from weighted homogeneity and sparse varieties.

Unification: It demonstrates that the local-global principle is robust for weighted homogeneous families, even when the branch locus is sparse, provided the geometric growth conditions are met.
Refinement of Density Theory: The work highlights that the "zeta exponent" governing visibility density is not solely a function of the ambient dimension or the variety's dimension, but is intrinsically linked to the growth rate of primitive points on the variety relative to the chosen height function. This allows for non-integral exponents (e.g., $3/2$ ) and zero densities.
Quantitative Improvement: The paper provides sharper quantitative bounds for the invisible loci in non-monomial polynomial families, improving upon previous work by Chaubey, Pandey, and Regavim.
Conjectures: The authors propose that for a broad class of visibility data with polynomial growth, the defect set always has density zero, and the global density is the product of local densities, even in sparse settings (Conjectures 1 and 2).

The authors explicitly state that their results are derived from the cited works (e.g., [CPR26], [GLM+03], [HKRS24]) and that the local-global failures in non-monomial cases are "density zero phenomena," consistent with the broader philosophy of strong approximation and thin-orbit problems.

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