Criterion for Complete Intersection of certain Monomial Curves

Let k be a field and m₀, …, m_e−1 (e ≥ 3) be a sequence of positive integers with gcd(m₀, …, m_e−1) = 1. Let <math display='block'> <mi mathvariant='script'>C</mi> </math>$$ \mathcal{C} $$ be the affine monomial curve in the e-space <math display='block'> <mrow> <msubsup> <mi mathvariant='double-struck'>A</mi> <mi>k</mi> <mi>e</mi> </msubsup> </mrow> </math>$$ \mathbb{A}_{k}^{e} $$ defined parametrically by <math display='block'> <mrow> <msup> <mi>T</mi> <mrow> <msub> <mi>m</mi> <mn>0</mn> </msub> </mrow> </msup> <mo>,</mo><mo>&#x2026;</mo><mo>,</mo><msub> <mi>X</mi> <mrow> <mi>e</mi><mo>&#x2212;</mo><mn>1</mn> </mrow> </msub> <mo>=</mo><msup> <mi>T</mi> <mrow> <msub> <mi>m</mi> <mrow> <mi>e</mi><mo>&#x2212;</mo><mn>1</mn> </mrow> </msub> </mrow> </msup> </mrow> </math> $${T^{{m_0}}}, \ldots ,{X_{e - 1}} = {T^{{m_{e - 1}}}}$$. In this article, we assume that m₀, …, m_e−1 form a minimal arithmetic sequence and find some criterion for complete intersection of C.