What is precise definition of "monomial curve" in affine $e$-space?

Question

What is precise definition of monomial curve in affine $e$-space ?

Answer 1

A curve is said to be monomial curve if $~\exists~$ a generator of its defining ideal which consists monomials only.

Answer 2

A monomial curve of degree $d$ with parameters $\mathcal{A}=\{0,a_1,\dots,a_c,d\}$ such that $0\lt a_1 \lt \dots \lt a_c \lt d$ is the curve defined by the parametrization

$$\mathbf{P}^1\rightarrow \mathbf{P}^{c+1}$$ $$(s,t)\mapsto(s^d,s^{d-a_1}t^{a_1},s^{d-a_2}t^{a_2},\dots,s^{d-a_c}t^{a_c},t^d)$$