Points below a parabola in affine planes of prime order

The elements of a finite field of prime order canonically correspond to the integers in an interval. This induces an ordering on the elements of the field. Using this ordering, Kiss and Somlai recently proved interesting properties of the set of points below the diagonal line. In this paper, we investigate the set of points lying below a parabola. We prove that in some sense, this set of points looks the same from all but two directions, despite having only one non-trivial automorphism. In addition, we study the sizes of these sets, and their intersection numbers with respect to lines.

💡 Research Summary

The paper studies the set of points lying below a quadratic curve in the affine plane AG(2, p) over a prime field Fₚ. For a non‑zero coefficient α∈Fₚ* and arbitrary β, γ∈Fₚ, define the quadratic polynomial f(x)=αx²+βx+γ and the point set

S = { (x, y)∈Fₚ² | y < f(x) }.

The authors introduce, for each direction d∈Fₚ∪{∞}, the projection function

pr_{S,d}(b) = |{ (x, y)∈S | y = d x + b }|

(the number of points of S on the line of slope d and intercept b; for d=∞ the line is x = –b).

Main results

1. Cyclic shift property (Theorem 3.1). Apart from the two distinguished directions (0) and (∞), all projection functions are cyclic shifts of each other: for any d∈Fₚ* and any b∈Fₚ

pr_{S,d}(b) = pr_{S,1}( b – β – d + ½ (d – ½α) ).

Consequently the image of each pr_{S,d} is an interval of consecutive integers. Using the Polya–Vinogradov inequality and bounds on the quadratic character χ, the length of this interval is shown to lie between √p /(2π) and √p log p.

2. Size of S (Theorem 3.3). The cardinality of S is always a multiple of p and satisfies

|S| = p²/2 ± cₚ p √p log p,

where the constant cₚ depends only on the congruence class of p:

• cₚ = 1 if p ≡ 1 (mod 4),
• cₚ = 2 if p ≡ –1 (mod 8),
• cₚ = 4/3 if p ≡ 3 (mod 8).

The proof rewrites |S| as Σ_{x∈Fₚ} ν(f(x)) (ν maps a field element to its integer representative in {0,…,p−1}) and then separates the contribution of squares and non‑squares, applying Lemma 2.6 on consecutive pairs of squares and the Polya–Vinogradov bound on sums involving χ.

3. Automorphism group (Theorem 3.5). For p ≥ 97, when AG(2, p) is embedded in the projective plane PG(2, p), the only non‑trivial projective linear transformation that stabilises S is the reflection (X, Y, Z) ↦ (−X, Y, Z). The proof distinguishes the two points (0) = (0 : 1 : 0) and (∞) = (1 : 0 : 0) from all other points by examining how many different intersection numbers lines through a given point can have with S; (0) and (∞) have uniquely large ranges because the image of f contains at least p − 5/4 consecutive field elements (Lemma 2.6). Once (0) and (∞) are fixed, any stabiliser must fix every line through (0) and consequently every line through (∞), forcing the transformation to be of the form (aX+bZ, Y, Z). Further constraints from the sizes of intersections force a = –1 and b = 0, yielding the claimed reflection.

4. Isomorphism classes (Theorem 3.7). Up to affine transformations, the sets S defined by different quadratics fall into exactly 2p non‑isomorphic families when β is normalized to 0 and α is either a square or a non‑square. The image of the projection in direction ∞ uniquely determines whether α is a square (image = (Q∪{0})+γ) or a non‑square (image = (N∪{0})+γ), and the value ν(γ) appears exactly once, allowing reconstruction of γ. Hence there are 2p distinct sets for the strict inequality y < f(x). Considering also the complementary sets defined by y ≤ f(x) and the possibility of shifting the inequality by 1 (which corresponds to taking complements), the total number of affine‑isomorphism classes is exactly 3p + 1.

Techniques and auxiliary results
The analysis relies heavily on classical results about quadratic characters: the Polya–Vinogradov inequality (Result 2.1) and its converse (Result 2.2), as well as precise counts of squares in intervals (Result 2.4) and bounds on the number of consecutive square pairs (Lemma 2.6). Lemma 2.6 guarantees that the image of any quadratic polynomial contains many consecutive field elements, a fact crucial for distinguishing the special directions (0) and (∞). The authors also use elementary linear changes of variables to normalize β to 0 and to show that only the quadratic character of α matters.

Significance
This work extends the recent study of “points below the diagonal” (Kiss–Somlai, 2024) to a genuinely quadratic setting. While the diagonal case yields exactly three special directions and a large automorphism group, the parabola case shows that, despite having only one non‑trivial symmetry, the point set looks essentially the same from all but two directions. The results illustrate how number‑theoretic estimates (quadratic character sums) can control geometric properties (directional distribution of points) in finite affine planes. The paper also opens avenues for studying higher‑degree curves or affine planes over non‑prime fields, where the interaction between character sums and combinatorial geometry may become even richer.