Resolution of an algebraic singularity by power geometry algorithms

Abstract

A polynomial in three variables is considered near a singular point where the polynomial itself and its partial derivatives vanish. A method for calculating asymptotic expansions in parameters for all branches of the set of roots of the polynomial near the singular point is proposed. The method is based on spatial power geometry and uses modern computer algebra algorithms: for calculating Gröbner bases and for work with algebraic curves. The implementation of the method is demonstrated on the example of a sixth degree poly nomial in three variables considered near infinity and near a degenerate singular point.

Unveiling regions in multi-scale Feynman integrals using singularities and power geometry

“…9. Using the theorem from Bruno [20] we conclude that the above truncated polynomials are satisfied by the following expressions for the alpha-parameters…”

“…Power geometry allows one to obtain solutions to polynomials and is particularly useful around singular points. Before we outline the procedure for obtaining solutions to polynomials using the techniques developed by Bruno [19,20,21], we briefly summarize the basic definitions and concepts which we have used in our subsequent analysis.…”

“…6. Finally, the vector P which gives us the scaling of the variables with respect to the parameter t is obtained, using the following theorem [20]:…”

Asymptotic Solving Essentially Nonlinear Problems

“…Implementation of the described algorithm see in [9]. Its application to computation of a set of stability of a certain ODE system depending on several parameters see in [10].…”

“…[11]) If y = c r x r , c = const, ω(1, r) ∈ U (d)j , is a power solution to truncated equation(9), then equation(7) has an expansion of solutions of the formz = c r x r + ∑ c s x s over s ∈ K(k 1 , . .…”