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If I have parametrized a curve in the complex plane by $$x(t)=a_lt^l+\cdots+a_nt^n$$ $$y(t)=b_kt^k+\cdots+b_mt^m$$ and the image is reduced (there exist at least two exponents which are relatively prime), how many blow ups do I need to perform at the singularity at $t=0$ (the point (0,0)) until my curve is smooth?

Any good references for figuring this out?

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