Suppose we have polynomials is of form $$h_1(x_1,\dots,x_{dn})-c_1\in\mathbb Z[x_1,\dots,x_{dn}]$$ $$\vdots$$ $$h_{nd}(x_1,\dots,x_{dn})-c_{nd}\in\mathbb Z[x_1,\dots,x_{dn}]$$ where $h_1(x_1,\dots,x_{dn}),\dots,h_n(x_1,\dots,x_{dn})\in\mathbb Z[x_1,\dots,x_{nd}]$ are homogeneous polynoials of degree $d$ with each $x_i$ degree $1$ (that is only monomials of form $x_{i_1},\dots,x_{i_d}$) on the conditions that

  1. each coefficient except $c_1,\dots,c_{nd}$ is random in $[-B,B]\setminus\{0\}$

  2. with $c_1,\dots,c_{nd}$ being of absolute value at most $B^{1+\frac1{2^d-1}}$

  3. in each monomial $x_{i_1},\dots,x_{i_d}$ we have $i_j\in\{(j-1)n+1,\dots,jn\}$.

Then if you use elimination theory we can successively eliminate variables and reduce to an univariate polynomial.

Let the absolute value of the maximum coefficient size $B'$ of final univariate polynomial in reduced form (no common gcd of coefficients) and the let the degree be $d'$.

Then can $B'^{\frac1{d'}+\epsilon}>B^{\frac1{d(2^d-1)}}$ hold with probability $1-o(1)$ at arbitrarily small $\epsilon>0$ as $B$ increases and $d$ is fixed?

  • $\begingroup$ Editing a question with high frequency indicates that the questioner hasn't thought enough about his/her question before asking. (And my experience is to better stay away from questions that are not well-thought.) $\endgroup$ – tj_ Jan 11 at 23:42

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