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If I were to present you with a general algebraic equation, something like a polynomial (involving just addition, multiplication, subtraction but also division) but with complex exponents, how many solutions would there be?

A simplish example would be:

$x^{2.5}-3*x^2+11*x^{1.25}-x^{-1}=0$

You could graph it, I suppose (even though it wouldn't be a simple cartesian graph if you're expecting complex outputs), but is there a general way to identify the number of roots or the shape of the graph? Is there a discriminant for these kinds of equations? To clarify, there would only ever be one equation and one variable.

I don't know much about this so any help whatsoever would be appreciated. Thanks

Edit: The example I gave was with all rational real powers- but I am looking to generalize this to complex and irrational indices, such as;

$x^{2.5+3i}-3*x^{\sqrt{π}}+11*x^{\frac97}-x^{-i}=0$.

I realise I may have not ordered that appropriately, but I don't really know how to.

MukundKS
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  • Is $x$ real or complex? – Arastas Nov 20 '20 at 09:10
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    If all powers are rational (as in your example) then you can rewrite it as an integer-degree polynomial by substitution. Check this one: https://math.stackexchange.com/questions/1291208/number-of-roots-of-a-polynomial-of-non-integer-degree – Arastas Nov 20 '20 at 09:12

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