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Largest linear independent subset of n polynomials

In the format below n :

x(i)  
x(i)+1  
x(i)+x(k)  
x(i)+x(k)+1  
...

We want to extract the largest possible linear independent subset of the above polynomials.
I tried to destroy Gassian, but it's too slow.
Any help is appreciated

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When I was looking, the solution is to use SVD or QR Decomposition.
This is true. Relations



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