In this paper, we introduce a new and direct approach to study the solvability of systems of equations generated by bilinear forms. More precisely, let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$B (\cdot , \cdot )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo>,</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a non-degenerate bilinear form and E be a set in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {F}_q^2$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="double-struck">F</mi> <mi>q</mi> <mn>2</mn> </msubsup> </math></EquationSource> </InlineEquation>. We prove that if <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$|E|\gg q^{5/3}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> <mi>E</mi> <mo stretchy="false">|</mo> </mrow> <mo>≫</mo> <msup> <mi>q</mi> <mrow> <mn>5</mn> <mo stretchy="false">/</mo> <mn>3</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> then the number of triples (B(x, y), B(y, z), B(z, x)) with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$x, y, z\in E$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mi>E</mi> </mrow> </math></EquationSource> </InlineEquation> is at least <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$cq^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <msup> <mi>q</mi> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> for some positive constant c. This significantly improves a result due to the fifth listed author (2009).

On the Solvability of Systems of Equations Revisited

期刊论文
发表日期：

2025年10月15日

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Thang Pham,
Steven Senger,
Nguyen Trung-Tuan,
Nguyen Duc-Thang,
Le Anh Vinh

摘要

In this paper, we introduce a new and direct approach to study the solvability of systems of equations generated by bilinear forms. More precisely, let $B (\cdot , \cdot )$ $B (\cdot, \cdot)$ be a non-degenerate bilinear form and E be a set in $\mathbb {F}_q^2$ $F_{q}^{2}$ . We prove that if $|E|\gg q^{5/3}$ $| E | ≫ q^{5 / 3}$ then the number of triples (B(x, y), B(y, z), B(z, x)) with $x, y, z\in E$ $x, y, z \in E$ is at least $cq^3$ $c q^{3}$ for some positive constant c. This significantly improves a result due to the fifth listed author (2009).