In general, how does one determine if a rational function

Casertowm7

Casertowm7

Answered question

2022-02-15

In general, how does one determine if a rational function is regular? I have the particular problem of determining in which points of the circle V(x2+y21)A2 is the rational function α=y1x regular?

Answer & Explanation

blokova5u8

blokova5u8

Beginner2022-02-16Added 7 answers

To say that ϕ=y1x is regular on the circle means that there exist polynomials p(X,Y),q(X,Y)ϵk[X,Y] such that:
Y1=Xp(X,Y)+q(X,Y)(x2+Y21)ϵk[X,Y]
But substituting X=0 in that equality yields
Y1=q(0,Y)(Y21)ϵk[Y]
which is impossible since the left hand side has degree 1 whereas the right hand side is zero or has degree 2. Hence the rational function ϕ is not regular.
Since ϕ is not regular but is clearly regular at all points of the circle different from P=(0,1), it follows that P is the only point where ϕ is not regular, i.e. the only pole of ϕ.

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