On the Pythagoras number of a real irreducible algebroid curve

Jesus J. Ortega

doi:10.1007/bf01446562

HIGHER PRODUCT PYTHAGORAS NUMBERS OF SKEW FIELDS

Asian-European Journal of Mathematics ◽

10.1142/s1793557110000131 ◽

2010 ◽

Vol 03 (01) ◽

pp. 193-207

Author(s):

Dejan Velušček

Keyword(s):

Laurent Series ◽

Skew Field ◽

Skew Fields ◽

Pythagoras Number

We introduce the n–th product Pythagoras number p n(D), the skew field analogue of the n–th Pythagoras number of a field. For a valued skew field (D, v) where v has the property of preserving sums of permuted products of n–th powers when passing to the residue skew field k v and where Newton's lemma holds for polynomials of the form Xn - a, a ∈ 1 + I v , p n(D) is bounded above by either p n( k v ) or p n( k v ) + 1. Spherical completeness of a valued skew field (D, v) implies that the Newton's lemma holds for Xn - a, a ∈ 1 + I v but the lemma does not hold for arbitrary polynomials. Using the above results we deduce that p n (D((G))) = p n(D) for skew fields of generalized Laurent series.

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A lattice of extension rings for a commutative ring

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100055067 ◽

1978 ◽

Vol 84 (2) ◽

pp. 225-234 ◽

Cited By ~ 2

Author(s):

D. Kirby ◽

M. R. Adranghi

Keyword(s):

Point Of View ◽

Local Rings ◽

Algebraic Point ◽

Multiple Points ◽

One Dimensional ◽

Maximal Ideals ◽

Blowing Up ◽

Abstract Case ◽

Algebroid Curve

The work of this note was motivated in the first place by North-cott's theory of dilatations for one-dimensional local rings (see, for example (4) and (5)). This produces a tree of local rings as in (4) which corresponds, in the abstract case, to the branching sequence of infinitely-near multiple points on an algebroid curve. From the algebraic point of view it seems more natural to characterize such one-dimensional local rings R by means of the set of rings which arise by blowing up all ideals Q which are primary for the maximal ideals M of R. This set of rings forms a lattice (R), ordered by inclusion, each ring S of which is a finite R-module. Moreover the length of the R-module S/R is just the reduction number of the corresponding ideal Q (cf. theorem 1 of Northcott (6)). Thus the lattice (R) provides a finer classification of the rings R than does the set of reduction numbers (cf. Kirby (1)).

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Analytic surface germs with minimal Pythagoras number

Mathematische Zeitschrift ◽

10.1007/s00209-005-0811-z ◽

2005 ◽

Vol 250 (4) ◽

pp. 967-969 ◽

Cited By ~ 1

Author(s):

José F. Fernando

Keyword(s):

Analytic Surface ◽

Pythagoras Number

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Algorithms for quadratic forms over global function fields of odd characteristic

ACM Communications in Computer Algebra ◽

10.1145/3511528.3511530 ◽

2021 ◽

Vol 55 (3) ◽

pp. 68-72

Author(s):

Mawunyo Kofi Darkey-Mensah

Keyword(s):

Quadratic Form ◽

Quadratic Forms ◽

Function Fields ◽

Number Fields ◽

Global Function ◽

Global Function Fields ◽

Anisotropic Part ◽

Pythagoras Number ◽

Present Algorithm

This paper presents an adaptation of recently developed algorithms for quadratic forms over number fields in [4] to global function fields of odd characteristics. First, we present algorithm for checking if a given non-degenerate quadratic form is isotropic or hyperbolic. Next we devise a method for computing the dimension of the anisotropic part of a quadratic form. Finally we present algorithms computing two field invariants: the level and the Pythagoras number.

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