Modular Resolution by Polyseries

We study the modular resolution method using a new tool called a Polyserie introduced by Wildberger N.J. & Rubine D. to prove an equivalence theorem of the existence and the unicity of the solution of the equation of the form <i>H<sub>t</sub><sup>2</sup>(x) – H<sub>t</sub>(x)+t ≡ 0 (mod t<sup>n</sup>)</i>. These equations admits solutions for a fixed <i>t</i> but when <i>t</i> changes in a range of several values, then the equation becomes parametric and hence the solution should be parametric too. Prof. Wildberger N.J has studied such equations using Catalan numbers sequence using the recurrence formula between their terms C<sub>n+1</sub> = ∑<sub>k=0</sub><sup>n</sup> C<sub>k</sub> C<sub>n-k</sub>, which allow us to easily prove the main result of the present article. We study Wildberger’s Polynumber Sequences, Binomial Chu-Vandermonde Identity, Truncated Polyseries and finally Modular Resolution as Application.