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I came across this notation while at this MathOverflow thread and I could not find its meaning. It makes the biggest sense that $f(x,y) \in \Bbb Q[x,y] $ represents any continuous function on the interval $[x,y] \in \Bbb Q$ and thus $\Bbb Q[x,y]$ is a set of such functions. However, I am not sure.

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2 Answers 2

up vote 7 down vote accepted

It means a polynomial in two variables with coefficients in the rational numbers $\mathbb Q.$

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More generally, $\Bbb Q[x,y]$ means the ring (field) of all bivariate polynomials with rational coefficients. –  Arkamis Aug 4 '12 at 23:58
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@EdGorcenski , not a field. –  Will Jagy Aug 5 '12 at 1:43
    
So if I understand, could the following be the examples of such polynomials? x + y, (2/3)*x + y^2, (1/2)*x + (3/4)*y, etc. –  David Toth Aug 5 '12 at 1:46
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@dt1510 yes, but also $ x^3 + x^2 y - (2/3)y^3 - x^2 + (1/4)x - y - (3/7) $ –  Will Jagy Aug 5 '12 at 1:55
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@WillJagy Of course, you are correct. Silly oversight on my part. I was thinking of the field $\Bbb Q$ and had wanted to say something more about it, and got my wires crossed :) Thanks for the correction. –  Arkamis Aug 5 '12 at 3:55

Or, it means the rationals adjoined with the elements x,y as a ring. So in @Will Jagy 's answer, x,y are indeterminants and therefore transcendentals. But x,y could also be other numbers that are not transcendental (i.e. algebraic), but you would need explicit values. For example, $x=\sqrt2$ and $y=\sqrt3$ would also work, but these are algebraic (i.e. solutions of polynomials over the rationals). This matters because it results in very different rings.

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Thank you for your answer too. –  David Toth Aug 5 '12 at 4:21

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