# Is there a variation of Coppersmith's method that applies to disjoint variable set with additional control?

We have a polynomial $$f(x_1,x_2,x_3,x_4)\in\mathbb Z[x_1,x_2,x_3,x_4]$$ where the only monomials are either from set $$\{x_1,x_1x_2,x_2,x_3,x_3x_4,x_4\}$$ and we seek solutions $$(x_1,x_2,x_3,x_4)\in\mathbb Z^4$$ with $$|x_1|, $$|x_2|, $$|x_3| and $$|x_4|. We see that $$x_1,x_2$$ and $$x_3,x_4$$ variables do not mix.

It seems the set of $$x_1,x_2$$ variables and set of $$x_3,x_4$$ variables each both individually and collectively satisfy the generalized lower triangle bound on page $$16$$ http://www.cits.rub.de/imperia/md/content/may/paper/jochemszmay.pdf with $$\lambda_1=\lambda_2=\lambda_3=\lambda_4=2$$ and $$D=1$$.

Assume we have the additional condition that for a given $$(x_1,x_2)\in\mathbb Z^2$$ with $$|x_1| and $$|x_2| we have that there is an unique $$(x_3,x_4)\in\mathbb Z^2$$ with $$f(x_1,x_2,x_3,x_4)=0$$, $$|x_3| and $$|x_4| vice versa for a given $$(x_3,x_4)\in\mathbb Z^2$$ with $$|x_3| and $$|x_4| we have that there is an unique $$(x_1,x_2)\in\mathbb Z^2$$ with $$f(x_1,x_2,x_3,x_4)=0$$, $$|x_1| and $$|x_2|. $$W$$ is highest absolute value of coefficient of $$f(x_1X_1,x_2X_2,x_3X_3,x_4X_4)$$.

1. In this situation can the bound of $$X_1^{\lambda_1}X_2^{\lambda_2}X_3^{\lambda_3}X_4^{\lambda_4}\leq W^\frac1D$$ be improved to $$\max(X_1^{\lambda_1}X_2^{\lambda_2},X_3^{\lambda_3}X_4^{\lambda_4})\leq W^\frac1D$$ or may be at least $$\max(X_1^{\lambda_1}X_2^{\lambda_2},X_3^{\lambda_3/2}X_4^{\lambda_4/2})\leq W^\frac2{3D}$$ ($$\lambda_3/2$$ and $$\lambda_4/2$$ is based on guess that variables are disjoint and have separate control and $$x_3,x_4$$ do not satisfy generalized triangle bound with $$\lambda_3=\lambda_4=1$$ and assuming $$x_1,x_2$$ variables were not present in given polynomial will give $$W^{\frac2{3D}}$$ bound)?

2. If not what is the best we can do at least for the case $$X_1=X_2=X_3=X_4$$?