Howgrave-Graham condition says roots of $$f(x_1,\dots,x_n)\equiv0\bmod R$$ at an $R\in\mathbb N$ are roots of $$f(x_1,\dots,x_n)=0$$ over $\mathbb Z$ if $$\|f(x_1X_1,\dots,x_nX_n)\|<\frac{R}{\sqrt{\#\mbox{ of monomials in }f}}$$ (here $|x_i|<X_i$ and $|y_j|<Y_j$ are root bounds constraints) is satisfied.

In typical applications $R=N$ is taken and we construct a lattice with $$(f(x_1,\dots,x_n))^jN^{m-j}\prod_{t=1}^n x_t^{i_t}$$ at an integer $m$ at suitable $i_1,\dots,i_n\in\mathbb N$. The shortest vector bounds of this lattice from LLL together with Howgrave-Graham bound indicates maximum allowable size of root bounds $X_i,Y_j$.

If instead of bound $$\|f(x_1X_1,\dots,x_nX_n)\|<\frac{R}{\sqrt{\#\mbox{ of monomials in }f}}$$ we satisfy $$\zeta_n\|f(x_1X_1,\dots,x_nX_n)\|<\frac{R}{\sqrt{\#\mbox{ of monomials in }f}}$$ at some $\zeta_n\in(0,1)$ then there is no guarantee there are common roots of polynomials obtained with independent vectors obtained from lattice with given polynomial all the time.

However what is the probability at which this happens we can guarantee common roots of polynomials with independent vectors obtained from lattice with given polynomial?


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