Inversion of many-dimensional Mellin transforms and solutions of algebraic equations

On Convergence of Mellin-Barnes Integrals Representing Solutions of General Algebraic Systems of 3 Equations with 3 Variables

“…By the proving above, the coefficient of t (2) in the second inequality and the coefficient of t (3) in the third inequality became are positive. Denote by…”

“…By conditions (3) the coefficients of this form are positive and the coordinates of non-zero vectors τ (1) and τ (2) are non-negative, so α 1 β 2 − α 2 β 1 > 0. Similarly, we prove the positiveness of all the remaining minors of the matrix from (8).…”

“…Further, the difference α 1 β 2 − α 2 β 1 = ⟨α (1) , τ (1) ⟩⟨β (2) , τ (2) ⟩ − ⟨α (2) , τ (2) ⟩⟨β (1) , τ (1) ⟩ is equal to bilinear form on τ (1) and τ (2) :…”

“…Take out t (k) from the sign of the scalar product ⟨α (1) , τ (1) ⟩t (1) + ⟨α (2) , τ (2) ⟩t (2) + ⟨α (3) , τ (3) ⟩t (3) , ⟨β (1) , τ (1) ⟩t (1) + ⟨β (2) , τ (2) ⟩t (2) + ⟨β (3) , τ (3) ⟩t (3) , ⟨γ (1) , τ (1) ⟩t (1) + ⟨γ (2) , τ (2) ⟩t (2) + ⟨γ (3) , τ (3) ⟩t (3) .…”

“…The full convergence domain was found in 2007 in the article of I. A. Antipova [2]. In [3] of V. A. Stepanenko received the integral of MellinBarnes type formally representing the solution of a general system of n polynomial equations with n variables (see also [4]).…”