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By Euler L.

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**Extra resources for An observation on the sums of divisors**

**Example text**

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 )=

### An observation on the sums of divisors by Euler L.

by Charles

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