Prime Representing Polynomial with 10 Unknowns – Introduction

The main purpose of the article is to construct a sophisticated polynomial proposed by Matiyasevich and Robinson [5] that is often used to reduce the number of unknowns in diophantine representations, using the Mizar [1], [2] formalism. The polynomial Jk(a1,…,ak,x)=∏ɛ1,…,ɛk∈{ ±1 }(x+ɛ1a1+ɛ2a2W)+…+ɛkakWk-1 {J_k}\left( {{a_1}, \ldots ,{a_k},x} \right) = \prod\limits_{{\varepsilon _1}, \ldots ,{\varepsilon _k} \in \left\{ { \pm 1} \right\}} {\left( {x + {\varepsilon _1}\sqrt {{a_1}} + {\varepsilon _2}\sqrt {{a_2}} W} \right) + \ldots + {\varepsilon _k}\sqrt {{a_k}} {W^{k - 1}}} with W=∑i=1kx i2 W = \sum\nolimits_{i = 1}^k {x_i^2} has integer coefficients and J_k(a₁, . . ., a_k, x) = 0 for some a₁, . . ., a_k, x ∈ ℤ if and only if a₁, . . ., a_k are all squares. However although it is nontrivial to observe that this expression is a polynomial, i.e., eliminating similar elements in the product of all combinations of signs we obtain an expression where every square root will occur with an even power. This work has been partially presented in [7].