Abstract: Let $S=\{x^2+c_1,\dots,x^2+c_s\}$ be a set of quadratic polynomials with integer coefficients and let $M_S$ be the semigroup generated by $S$ under composition. We prove that if $S$ contains at least two irreducible polynomials, then $M_S$ contains a large, explicit subset of irreducible polynomials. Moreover, if one wishes to apply the standard irreducibility test in this setting (looking for squares in a critical orbit), then our result is sharp. To do this, we use tools from arithmetic dynamics and Diophantine geometry.
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