What question did this study set out to answer?

To develop a higher-degree analogy of the Szemerédi–Trotter theorem in the context of finite fields and polynomial expansion.

February 24, 2026Proceedings of the Royal Society of Edinburgh Section A MathematicsOpen Access

Incidence geometry and polynomial expansion over finite fields

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To develop a higher-degree analogy of the Szemerédi–Trotter theorem in the context of finite fields and polynomial expansion.
Utilized spectral theory methods for the analysis.
Applied concepts from algebraic geometry in findings.
Focused on establishing relationships in finite fields.
Confirmed the existence of a higher-degree Szemerédi–Trotter-type theorem.
Illustrated the application of this theorem in polynomial expansion scenarios.

Implication

This analysis establishes a higher-degree analogue for polynomial expansion in finite fields, suggesting new applications in algebraic geometry.

Abstract

We use spectral theory and algebraic geometry to establish a higher-degree analogue of a Szemerédi–Trotter-type theorem over finite fields, with an application to polynomial expansion.

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Cite This Study

Arala et al. (2026) studied this question.

synapsesocial.com/papers/699d3ff8de8e28729cf64d53 https://doi.org/10.1017/prm.2026.10130

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Implication

This analysis establishes a higher-degree analogue for polynomial expansion in finite fields, suggesting new applications in algebraic geometry.

Abstract

We use spectral theory and algebraic geometry to establish a higher-degree analogue of a Szemerédi–Trotter-type theorem over finite fields, with an application to polynomial expansion.

Incidence geometry and polynomial expansion over finite fields

Key Points

Implication

Abstract

Discussion

Cite This Study

Incidence geometry and polynomial expansion over finite fields

Key Points

Implication

Abstract

Discussion

Cite This Study

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