Which is the correct definition of degree of a curve

Question

I found those definitions of what a degree is and they contradict each other so I don't know which is the correct. The first definition it was by the last year professor saying that the degree of an affine curve is any polynomial with no repeated factors which defines it.
But the professsor this year told us that if the affine curve defines a point of multiplicity d then its degree is d. But using the definition of the last year professor it should be 1.
So which one is the correct definition?
Thanks in advance

Answer 1

None of these definitions is correct: an affine curve does not have a degree !

The reason is that linearly equivalent divisors on such a curve don't have the same degree: for example given three distinct points $a,b,c$ on the affine line $\mathbb A^1$, the divisors $a+b$ and $c$ are linearly equivalent since $a+b=c+\operatorname {div} \frac {(z-a)(z-b)}{z-c}$, but those divisors have different degrees, namely $2$ and $1$.
By contrast two linearly equivalent divisors have the same degree on a smooth projective curve.
This is the deep reason for the existence of the degree $d$ of such a curve: $d$ is the degree of the divisor of intersection of the curve with an arbitrary hyperplane.
This definition makes sense because all these divisors of intersection are linearly equivalent and thus have the same degree.