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Is there any criterion for checking algebraic independence of a set of polynomials in $n$ variables in terms of the leading monomials with respect to some monomial order ? The Jacobian criterion is too hard to verify if the set of polynomials are not so simple. For example it was difficult for me to just verify the Jacobian criterion for the following set of simple looking polynomials. Is there any other way to check the algebraic independence in this case ?

$f_1=(z+t)(yz+yt+t)(xyz+xyt+xt+t)t^3$

$f_2=(z+t)(yz+yt+t)(xy+x+a)t^5$

$f_3=(z+t)(y+b)(xy+x+a)t^6$

$f_4=(y+b)(xy+x+a)t^7$

where $a,b$ are constants.

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