Analytic combinatorics in several variables (ACSV) generalizes the coefficient extraction of generating functions in one variable to several variables. Current developments in ACSV mostly concern rational or meromorphic generating functions by first representing coefficients via the multivariate Cauchy integral formula and then using Morse-theoretic homology arguments to deform the integral chain so that the integral becomes a sum of saddle point integrals. Coefficient asymptotics are previously known in the case when critical points of the Morse function are smooth points [PW02], multiple points [PW04, BMP24b], and quadratic cone points [BP11]. We generalize the result for multiple points to pseudo multiple points and show that these two kinds of points are similar under some conditions. The complexity hierarchy of ACSV goes up from rational functions to algebraic functions. By embedding the coefficient for an algebraic generating function as an elementary diagonal of a rational generating function with one more variable, [GMRW22] shows that the problem can be reduced to the well-known case of rational generating functions. We take a different approach, by lifting the torus in the Cauchy integral formula to the surface of the defining polynomial of the algebraic function, taking advantage of the covering space property of the surface. This leads to a similar computation to [GMRW22], avoids the Morse-theoretic homology arguments, and brings brighter transparency.