Ordered fields.
Ordered fields and sign-changing polynomials.
Ordered fields and the ultrafilter theorem
We prove that on the basis of ZF the ultrafilter theorem and the theorem of Artin-Schreier are equivalent. The latter says that every formally real field admits a total order.
Ordered Rings and Fields
We introduce ordered rings and fields following Artin-Schreier’s approach using positive cones. We show that such orderings coincide with total order relations and give examples of ordered (and non ordered) rings and fields. In particular we show that polynomial rings can be ordered in (at least) two different ways [8, 5, 4, 9]. This is the continuation of the development of algebraic hierarchy in Mizar [2, 3].
Ordering Epic R-Fields
Orderings under field extensions.
Ordre de transcendance sur
Ordres faibles et localisation de zéros de polynômes
Orthonormal bases for spaces of continuous and continuously differentiable functions defined on a subset of Zp.
Let K be a non-Archimedean valued field which contains Qp, and suppose that K is complete for the valuation |·|, which extends the p-adic valuation. Vq is the closure of the set {aqn | n = 0,1,2,...} where a and q are two units of Zp, q not a root of unity. C(Vq --> K) (resp. C1(Vq --> K)) is the Banach space of continuous functions (resp. continuously differentiable functions) from Vq to K. Our aim is to find orthonormal bases for C(Vq --> K) and C1(Vq --> K).
Ostrwoski's theorem for Henselian valued skew fields.
-adic difference-difference Lotka-Volterra equation and ultra-discrete limit.
-adic Siegel halfspace
-adic Teichmüller space and Siegel halfspace
-adic Teichmuller space for genus
P-adic root isolation.
We present an implemented algorithmic method for counting and isolating all p-adic roots of univariate polynomials f over the rational numbers. The roots of f are uniquely described by p-adic isolating balls, that can be refined to any desired precision; their p-adic distances are also computed precisely. The method is polynomial space in all input data including the prime p. We also investigate the uniformity of the method with respect to the coefficients of f and the primes p. Our method thus...
Pathological functions on Puiseux series ordered fields and others.
p-Henselian fields.
Polynômes eulériens modulo
Pour l'étude des groupes de ramification
Primideale im graduierten Wittring und im mod 2 Cohomologiering.