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Let $f,g \in \mathbb{C}[x,y]$. A comment to this question says that it is possible to write $g=qf+r$, for some $q,r \in k[x,y]$, but not necessarily with $\deg(r) < \deg(f)$.

Is there something interesting that can be said about $r$?

Moreover, its first answer says: "But all Euclidean is not lost, since one can generalize the polynomial division algorithm in a way that recovers many of the important properties".

Which properties?

In particular, we know that if $\gcd(f,g)=1$, then it is not true that necessarily there exist $u,v \in k[x,y]$ such that $uf+vg=1$, as for example $f=x$, $g=y$ show.

Thank you very much!

user237522
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    You might want to look up the notion of "monomial orders" which are often covered at the beginning of treatments of Groebner bases. – Daniel Schepler Feb 12 '20 at 00:01
  • @DanielSchepler, thank you; I guess you are right. Additional coments are welcome. (Is there a recommended book on the subject?). – user237522 Feb 12 '20 at 00:04
  • The most fundamental property of a Gröbner basis is that we can determine ideal membership using the division algorithm. If $I \subseteq k[x_1, \ldots, x_n]$ is an ideal and $g_1, \ldots, g_m$ is a Gröbner basis for $I$, then $f \in I$ if and only if its remainder $r$ upon division by $g_1, \ldots, g_m$ is $0$. Good references for Gröbner bases are Cox, Little, and O'Shea's Ideals, Varieties, and Algorithms and Ch. 9 of Dummit and Foote. – Viktor Vaughn Feb 12 '20 at 00:06
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    Umm, I think I first learned about Groebner bases from Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry which I found readable enough. I don't know that much about other references which might be heavier on calculation examples. – Daniel Schepler Feb 12 '20 at 00:07
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    Thank you very much both of you. – user237522 Feb 12 '20 at 00:09
  • @André3000, the property you mentiond sounds nice. I should try to learn that subject. – user237522 Feb 12 '20 at 00:15
  • Oh, indeed, it is very nice. I like to think of a Groebner basis as a set of polynomial equations with "no hidden surprises". And the primary purpose of Buchberger's algorithm is to generate all the surprising consequences of a starting set of polynomial equations - with the meta surprise being that a very limited algorithm for checking for such surprises eventually finds all the surprises that there are. – Daniel Schepler Feb 12 '20 at 00:25
  • @DanielSchepler, thank you for your explanation. (I am trying to read now wikipedia's article on the subject. Later I will check the references you and Andre recommended). – user237522 Feb 12 '20 at 00:34

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