Bézout's theorem intuition

Question

I am not taking algebraic geometry or trying to prove anything. I'm just looking for a simple intuitive understanding of Bézout's theorem for the case when one of the curves is a constant function. From https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem#Examples we have the example: "Two distinct non-parallel lines (in the same plane) always meet in exactly one point. Two parallel lines intersect at a unique point that lies at infinity."

But take y = 1 and y = x. These two lines meet once at (1,1), but the product of the degrees of these curves is 0*1 = 0.

I feel like I'm missing something very simple or don't understand the hypotheses of the theorem... can someone please help explain? Thanks!

Answer 1

The line described by $y=1$ is the zero set of a certain polynomial in the two variables $x$ and $y$, namely $y-1$. This polynomial has degree $1$, which is the exponent of $y$ in the first term. So, by definition, the degree of that curve is $1$.

On the other hand, the line described by $y=1$ is also the graph of a certain polynomial in the single variable $x$, namely $1$. This polynomial has degree $0$. But that isn't the polynomial relevant to Bézout's theorem!

Answer 2

The equation $y=1$ is linear, as is any equation $$ax+by+c=0,$$ where $a,b,c$ are constants. These are all of first degree since the operations on the variables involve nothing more than scalar multiplication and addition.