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It is believed [ 13 ,  19 ,  Section 4.1 ] that, among the coefficients entering Kontsevich’s formality quasi-isomorphism [ 18 ], there are irrational (possibly even transcendental) numbers. In this article, we prove that a formality quasi-isomorphism for Hochschild cochains of a polynomial algebra over   ${\mathbb Q}$   can be constructed recursively. The proof that the proposed recursive algorithm works, is based on the existence of formality quasi-isomorphism over   ${\mathbb R}$  . However, the algorithm requires no explicit knowledge of the coefficients entering Kontsevich’s construction. Although this algorithm completely bypasses Tamarkin’s approach [ 14 ,  23 ], the construction is inspired by Proposition 5.8 from classical paper [ 12 ] by V. Drinfeld.

A Formality Quasi-Isomorphism for Hochschild Cochains Over Rationals can be Constructed Recursively