<p>In this note we establish an upper bound for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$t \in \mathbb {Z} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </math></EquationSource> </InlineEquation> such that the vector <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(a_1+t, a_2 +t, \ldots , a_n+t) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>+</mo> <mi>t</mi> <mo>,</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo>+</mo> <mi>t</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mi>n</mi> </msub> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is multiplicative dependent for a fixed vector <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ (a_1, a_2, \ldots , a_n) \in {\mathbb {Z}^*}^n.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>∗</mo> </msup> </mrow> <mi>n</mi> </msup> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Additionally, we provide a family of values of <i>t</i> such that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(a_1+t,a_2+t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>+</mo> <mi>t</mi> <mo>,</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is multiplicative independent for a fixed multiplicative dependent vector <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(a_1,a_2).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Furthermore, we investigate the multiplicative dependent vectors <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(a_1,a_2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> such that the vectors <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(a_1+ a_2, a_1 - a_2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are also multiplicative dependent.</p>

Multiplicative dependence of translated vectors with integer coordinates

期刊论文
发表日期：

2025年12月12日

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Mitashree Behera,
Prasanta Kumar Ray

摘要

In this note we establish an upper bound for $t \in \mathbb {Z} $ $t \in Z$ such that the vector $(a_1+t, a_2 +t, \ldots , a_n+t) $ $(a_{1} + t, a_{2} + t, \dots, a_{n} + t)$ is multiplicative dependent for a fixed vector $ (a_1, a_2, \ldots , a_n) \in {\mathbb {Z}^*}^n.$ $(a_{1}, a_{2}, \dots, a_{n}) \in {Z^{*}}^{n} .$ Additionally, we provide a family of values of t such that $(a_1+t,a_2+t)$ $(a_{1} + t, a_{2} + t)$ is multiplicative independent for a fixed multiplicative dependent vector $(a_1,a_2).$ $(a_{1}, a_{2}) .$ Furthermore, we investigate the multiplicative dependent vectors $(a_1,a_2)$ $(a_{1}, a_{2})$ such that the vectors $(a_1+ a_2, a_1 - a_2)$ $(a_{1} + a_{2}, a_{1} - a_{2})$ are also multiplicative dependent.