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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.

Dave
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    I'd read it as "the polynomials which vanish on the line $y=0$". Note that those are indeed generated by the given monomials. Thus, I'd read "droite" as a convenient shorthand for "ligne droite", or "straight line". – lulu Sep 09 '19 at 20:23
  • Ah thank you, the argument is clear to me now! – Dave Sep 09 '19 at 20:27
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    I think I remember seeing, dually, a few 'lignes gauches' in some very old french math, referring to non-linear curves. – Badam Baplan Sep 09 '19 at 22:10
  • Thanks for the tip, I'm going to try and get some more practice reading mathematical French so it's useful to know – Dave Sep 09 '19 at 22:28

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It seems that "droite" means line here, so it's referring to polynomials that vanish on the line defined by the equation $y=0$.

Eric Wofsey
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