Shinji Miura gave certain multivariable polynomials that
express an Affine curve for a given algebraic function field
$F$ and its degree one place $\mathcal{O}$, if $F$ contains
such an $\mathcal{O}$. Suppose the equations contain $t$ ($\geq
2$) variables, and that the pole orders at $\mathcal{O}$ are
$a_1,\ldots,a_t \geq 1$, where $\operatorname{GCD}\{a_1,\ldots,a_t\}=1$.
If \[ \frac{a_i}{d_i} \in \frac{a_1}{d_{i-1}}{\mathbb
N}+ \cdots +\frac{a_{i-1}}{d_{i-1}}\mathbb{N}, \quad d_i=\operatorname{GCD}\{a_1,\ldots,a_{i}\}
\] for each $i=2,\ldots,t$, by arranging $a_1,\ldots,a_{t}$,
then we say that the orders $a_1,\ldots,a_{t}$ are telescopic.
On the other hand, the number $t'$ ($\geq t-1$) of the equations
in the Miura canonical form is determined by $a_1,\ldots,a_{t}$.
If $t'=t-1$, then we say that $a_1,\ldots,a_{t}$ are complete
intersection. It is known that the telescopic condition implies
the complete intersection condition. However, the converse
was open thus far. This paper solves the conjecture in the
affirmative by giving its proof.