I've been trying to read a proof linked in an answer to this question. It is written in French which unfortunately I do not speak, however I think I can follow most of the statements apart from one sentence, which I have highlighted in bold:
L’anneau $\Gamma(U)$ est donc isomorphe à l’anneau $R$ des polynômes $f(x,y)\in k[x,y]$ tels que $f(x,0)$ est constant. Ces polynômes sont exactement ceux de la forme $c+g(x,y)$ avec $c\in k$ et $g(x,y)\in yk[x,y]$. Comme $k$-algèbre $R$ est engendré par les monômes $x^my^{1+n}$ où $m,n\geq0$. L’idéal $I\subset R$ des polynômes qui s’anullent sur la droite $y=0$ est engendré par ces mêmes monômes.
I hope this is enough context for those who might not want to follow the link.
It is the meaning of "s’anullent sur la droite $y=0$" that I'm unsure about. As far as I can tell, the sentence says something like "The ideal $I\subset R$ of polynomials that cancel on the right $y=0$ is generated by the same monomials", but I'm struggling to interpret this.
It feels like I'm missing something between "on the right" and "$y=0$". I'm also not certain which monomials it is referring to.
If anyone thinks this question would be more appropriate for somewhere such as the French Language Stack Exchange I'm happy to move it, but the specific mathematical context seems quite important and so I have posted it here.
Any help would be much appreciated.
It seems that "droite" means line here, so it's referring to polynomials that vanish on the line defined by the equation $y=0$.