Breaking the symmetry in excess intersection and counting solutions of systems of polynomials

We revisit the fundamental problem of assigning intersection multiplicities to subsets of solutions of (square) systems of polynomials. Severi [Ann. Mat. Pura Appl. 26 (4), 1947] suggested an intuitive dynamic solution to this problem which was later corrected and made rigorous by Lazarsfeld [Compos. Math. 43, 1981]. We consider an asymmetric variant of this approach and find an explicit description of the resulting "ordered intersection multiplicity" which opens pathways to step by step solutions to the affine B\'ezout problem of counting isolated solutions to (square) systems of polynomials via "Bernstein-Kushnirenko type" estimates in terms of Newton diagrams. To illustrate our methods we compute the number of common tangent lines to $4$ general spheres in the affine $3$-space (which is known to be 12 due to Macdonald, Pach, and Theobald [Discrete Comput. Geom. 26 (1), 2001]) via certain ordered intersection multiplicities on the corresponding Grassmannian.