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Using Coppersmith for a second trivariate polynomial

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I have a trivariate polynomial whose roots I am interested. The polynomial has monomials in $\{X^4,X^2,X^2Y,X^2Z,1\}$. What is the best way to generate the lattice and apply $LLL$ so that I can get a second polynomial in common with the first under the assumption the polynomial has size of coefficients which satisfy Howgrave-Graham bound applicable to Coppersmith's techniques?

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