, Ao = A.integer_ring(
, Ao Out: Q 2 {x, } ? We can now create and manipulate elements: In: f = 2*x^2 + 5*x*y^2 g = 4 + 2*x^2*y f + g Out: ...00101x 2 + ...000010x 2 + ...000010x 2 + ...0000100 In: (1+g)
, The big-oh appearing on the last line hides terms which are multiple of 2 5, We observe that, in the outputs, terms are ordered with respect to the term order on T {X}, the greatest one coming rst
, Ideals of K {X} can be dened and manipulated as follows
, J
, *g in J Out: True In: log(1+g) in J Out: True And similarly for ideals of K {X} ? (observe that no losses of precision occur this time
000010x 2 + ...000100 + O (2 6 Q 2 {x ,
An Introduction to Gröbner Bases, Amer. Math. Soc, vol.7, 1994. ,
Non-Archimedean analysis, 1984. ,
Ein Algorithmus zum Aunden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal (An Algorithm for Finding the Basis Elements in the Residue Class Ring Modulo a Zero Dimensional Polynomial Ideal), English translation in J. of Symbolic Computation, Special Issue on Logic, Mathematics, and Computer Science: Interactions, vol.41, pp.475-511, 2006. ,
Linear Algebra over Z p, LMS J. Comput. Math, vol.17, pp.302-344, 2014. ,
Gröbner bases over elds with valuations, Math. Comp, vol.88, pp.467-483, 2019. ,
Ideals, varieties, and algorithms, Undergraduate Texts in Mathematics, 2015. ,
A new ecient algorithm for computing Gröbner bases (F4), Journal of Pure and Applied Algebra, 1999. ,
Algorithms in Local Algebra, Journal of Symbolic Computation, vol.19, pp.545-557, 1995. ,
An Algorithm for Computing a Gröbner Basis of a Polynomial Ideal over a Ring with Zero Divisors, Mathematics in Computer Science, 2009. ,
An algorithm to compute the equations of tangent cones, Proceedings of European Computer Algebra Conference in Marseille, pp.158-165, 1982. ,
, The Sage Development Team, SageMath, the Sage Mathematics Software System (Version 8.6), 2018.
Strong Grobner bases and cyclic codes over a nite-chain ring. Electronic notes in discrete maths 6, pp.240-250, 2001. ,
Rigid analytic spaces, Inventiones Mathematicae, vol.12, pp.257-289, 1971. ,
Matrix-F5 algorithms over nite-precision complete discrete valuation elds, Proceedings of 39th International Symposium on Symbolic and Algebraic Computation, ISSAC'14 ,
Précision p-adique ,
Matrix-F5 Algorithms and Tropical Gröbner Bases Computation, Proceedings of the 40th International Symposium on Symbolic and Algebraic Computation, 2015. ,
A Tropical F5 algorithm, Proceedings of the 42th International Symposium on Symbolic and Algebraic Computation, 2017. ,
On Ane Tropical F5 algorithm, Proceedings of the 43th International Symposium on Symbolic and Algebraic Computation, 2018. ,