Testing linear dependence of hyperexponential elements

Abstract

A Wronskian (resp. Casoratian) criterion is useful to test linear dependence of elements in a differential (resp. difference) field over constants. We generalize this criterion for invertible hyperexponential elements in a differential-difference ring extension over a field F. The generalization also enables us to connect similarity and F-linear dependence of invertible hyperexponential elements.

On the structure of compatible rational functions

“…Set b = m j=1 bjxj. Then a − b is in L1 + F(t, x1, y) and L1 +F(t, xm, y) by (14), (15) and the claim. Thus, a−b is in L1 + F(t, y) by the first assertion (setting Z1 = {x1}, Z2 = {xm}, and A = F(t2, .…”

“…Let a∈N1,1 ∩N1,m. Then (15) where f,f ∈ F(t, x, y), gj, r ∈ F(t, x1, y),gj,r ∈ F(t, xm, y) and ff gjgj = 0. For all j with 1 ≤ j ≤ m, let Pj be the polynomial part of a w.r.t.…”

“…a ≡ a1 mod Li and a ≡ a2 mod Li. To prove the second assertion, assume i = 1, d = 1 and e = m. Note that (15) where f, f ∈ F(t, x, y), gj, r ∈ F(t, x1, y), gj, r ∈ F(t, xm, y) and f f gj gj = 0. For all j with 1 ≤ j ≤ m, let Pj be the polynomial part of a w.r.t.…”

“…Proof. It follows from [15,Corollary 4.2] that h1, · · · , hs are algebraically dependent over F(t, x, y) iff there exist integers ω1, . .…”

Parallel telescoping and parameterized Picard-Vessiot theory

“…In fact, s i can be any integer between 1 and ℓ and w i,si must be zero if T (h i ) = 0; and s i is unique if T (h i ) is nonzero by Proposition 4.1 in [18]. Thus, (11) can be rewritten as…”