Computing with quadratic forms over number fields

Koprowski, Przemysław; Czogała, Alfred

doi:10.1016/j.jsc.2017.11.009

Cited by 14 publications

(11 citation statements)

References 14 publications

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“…Remark 6. In the next algorithms, Algorithms 3, 4, 5 and 6, we perform two kinds of factorization in like manner as in [7]. The first one is to find all dyadic primes of a given field, i.e.…”

Section: Algorithm 1: Length In a Non-dyadic Completionmentioning

confidence: 99%

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Computing the Length of Sum of Squares and Pythagoras Element in a Global Field

Darkey-Mensah,

Rothkegel

2021

Preprint

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This paper presents algorithms for computing the length of a sum of squares and a Pythagoras element in a global field K of characteristic different from 2. In the first part of the paper, we present algorithms for computing the length in a non-dyadic and dyadic (if K is a number field) completion of K. These two algorithms serve as subsidiary steps for computing lengths in global fields. In the second part of the paper we present a procedure for constructing an element whose length equals the Pythagoras number of a global field, termed a Pythagoras element.

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Section: Algorithm 1: Length In a Non-dyadic Completionmentioning

confidence: 99%

“…In this paper, we pursue the continuation of the recent work by P. Koprowski and A. Czoga la in [7] on the computational aspects of the theory of quadratic forms over global fields. In [7] the authors focused on algorithms over number fields (i.e. finite extensions of Q).…”

Section: Introductionmentioning

confidence: 99%

Computing the Length of Sum of Squares and Pythagoras Element in a Global Field

Darkey-Mensah,

Rothkegel

2021

Preprint

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“…We used scripts [15] written for PARI/GP [19] to test for which knots this lower bound would be applicable, finding the six knots: For these knots, the topological slice genus is at least 2, and, by the upper bounds already known, in fact equal to 2. For all of them, one takes the odd prime p in Theorem 1 to be 3.…”

Section: Taylor's Lower Boundmentioning

confidence: 99%

On Calculating the Slice Genera of 11- and 12-Crossing Knots

Lewark

McCoy

2017

Experimental Mathematics

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This paper contains the results of efforts to determine values of the smooth and the topological slice genus of 11-and 12-crossing knots. Upper bounds for these genera were produced by using a computer to search for genus one concordances between knots. For the topological slice genus further upper bounds were produced using the algebraic genus. Lower bounds were obtained using a new obstruction from the Seifert form and by use of Donaldson's diagonalization theorem. These results complete the computation of the topological slice genera for all knots with at most 11 crossings and leaves the smooth genera unknown for only two 11-crossing knots. For 12 crossings there remain merely 25 knots whose smooth or topological slice genus is unknown.

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“…The first two problems are simple, since they can be solved checking some easy to compute local invariants and applying the local-global principle (see e.g., [14,Section VI.3]). For details we refer the reader to [13,Algorithms 5 and 9]. The two remaining tasks present a different level of difficulty.…”

Section: Introductionmentioning

confidence: 99%