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In the formulation of the algorithm the direct sum of formal power series E suffices to give a notion of solutions coherent to the algebraic case: For e ∈ E we define the F -algebra homomorphism (j) φe : F {U } → F : ui → α(e)(u(j) ) evaluating all differential variables of a differential polynomial at the power series e. A differential equation or inequation for m functions U = {u(1) , . . , u(m) } in n indeterminates is an element p ∈ F {U } written p= or p= , respectively. A Thomas Decomposition of Algebraic and Differential Systems 43 solution of p= or p= is an e ∈ E with φe (p) = 0 or φe (p) = 0, respectively.

Furthermore rank(q) < rank((ST )x ). t. t. S. t. S. Algorithm: 1: (i, S1 , S2 ) ← ResSplit (S, (ST )x , q) 2: (S2 )Q ← (S2 )Q ∪ {q} 3: return S1 , S2 , PRSi ((ST )x , q, x)= Thomas Decomposition of Algebraic and Differential Systems 39 The following algorithm is similar, but instead of the gcd, it returns the first input polynomial divided by the gcd. It is used to assimilate an inequation into a system where there already is an equation with the same leader, or to calculate the least common multiple of two inequations.

The result of the Reduce algorithm does not need to be a canonical normal form. It only needs to detect polynomials that vanish on all solutions of a system: 1 In our context prem does not necessarily have to be the classical pseudo remainder, but any sparse pseudo remainder with property (1) will suffice. 36 T. Bächler et al. 6. Let p ∈ R with ld(p) = x. Reduce(S, p) = 0 implies φa (p) = 0 ∀ a ∈ Sol(S≤x ). , (ST )≤x is simple. If it is not simple, but ld(p) = x and (SQ )=

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An observation on the sums of divisors by Euler L.

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