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Waring's problem for fields

William Ellison (2013)

Acta Arithmetica

If K is a field, denote by P(K,k) the a ∈ K which are sums of kth powers of elements of K, by P⁺(K,k) the set of a ∈ K which are sums of kth powers of totally positive elements of K. We give some simple conditions for which there exist integers w(K,k) and g(K,k) such that: a ∈ P(K,k) implies that a is the sum of at most w(K,k) kth powers; a ∈ P⁺(K,k) implies that a is the sum of at most g(K,k) totally positive kth powers. We apply the results to characterise functions that are sums of kth powers...

Waring's problem for sixteen biquadrates. Numerical results

Jean-Marc Deshouillers, François Hennecart, Bernard Landreau (2000)

Journal de théorie des nombres de Bordeaux

We explain the algorithms that we have implemented to show that all integers congruent to 4 modulo 80 in the interval [ 6 × 10 12 ; 2 . 17 × 10 14 ] are sums of five fourth powers, and that all integers congruent to 6 , 21 or 36 modulo 80 in the interval [ 6 × 10 12 ; 1 . 36 × 10 23 ] are sums of seven fourth powers. We also give some results related to small sums of biquadrates. Combining with the Dickson ascent method, we deduce that all integers in the interval [ 13793 ; 10 245 ] are sums of 16 biquadrates.

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