M. Kontsevich and Y. Soibelman [“Stability structures, motivic Donaldson-Thomas invariants and cluster transformations”, arXiv:0811.2435; see also Contemp. Math. 527, 55–89 (2010; Zbl 1214.14014)]) made the following conjecture. Let \(F\) be a formal series on affine space \({\mathbb A}^{d_1} \times {\mathbb A}^{d_2} \times {\mathbb A}^{d_3}\), depending in a constructible way on finitely many extra parameters, such that \(F\) has degree zero with respect to the diagonal action of the multiplicative group \({\mathbb G}_m\) with weights \((1,-1,0)\). In particular, \(F(x,0,0)\equiv 0\). Let \(i_1: {\mathbb A}^{d_1}\times {\mathbb G}_m\to V(F)\times {\mathbb G}_m\) and \(i_0: \{0\}\times {\mathbb G}_m\to V(F)\times {\mathbb G}_m\) be the natural inclusions. Write \(F(0,0,z)=h(z)\). Then the following formula should hold: \[ \int_{{\mathbb A}^{d_1}}i_1^*{\mathcal S}_F={\mathbb L}^{d_1}i_0^*{\mathcal S}_{h}\;, \] where \({\mathcal S}_F\) and \({\mathcal S}_h\) are the motivic Milnor fibre of \(F\) and \(h\).
In this paper this formula is proved in some special cases, when \(F(x,y,z)=f(g_1(x,y),g_2(z)\) is a composition of a polynomial in two variables and a pair of regular functions, or \(F\) has the form \(F(x,y,z)=g(x,y,z)+h(z)^l\) with \(l\) sufficiently large under some additional nondegeneracy conditions. The author uses the explicit computation of the motivic Milnor fibres of a regular function using its Newton polyhedron.