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Let $\lambda \in \mathbb{Z^+}$ be a constant such that the polynomials $f(x,y)=4x^{10+\lambda }+7y^{21+\lambda }-1$ and $g\left(x,y\right)=5x^8+9y^{21}-1$ have common roots that belong to $\mathbb{R}$. Is there a theorem using which the solutions to the polynomial equations can be found?

I have tried using the gcd of the polynomials via wolfram, the answer was 1. That doesn't make sense because some of the next questions I have take that there are common roots as granted... I have tried using Groebner bases but failed, and U also tried converting the system to a linear one and thus using matrices. That was a desperate try and doesn't make sense to me, but still did it.

user26857
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pieq3
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    https://math.stackexchange.com/questions/4385271/find-lambda-in-mathbbn-such-that-4x11-lambda7y19-lambda-1-5x6 is similar but powers are different – cineel Feb 22 '22 at 18:20
  • @Greg The system will always have complex roots. If you are only looking for a certain type of roots (integer, real etc) then you should state so in the question. – dxiv Feb 22 '22 at 21:56
  • @dxiv Yes ,you are absolutely right , i was looking for real roots,fixed it. – pieq3 Feb 23 '22 at 13:45

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