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Questions tagged [groebner-basis]

A particular kind of generating subset of an ideal in a polynomial ring

A Gröbner basis is a particular kind of generating subset of an ideal I in a polynomial ring R. One can view it as a multivariate, non-linear generalization of:

  • the Euclidean algorithm for computation of univariate greatest common divisors,
  • Gaussian elimination for linear systems, and
  • integer programming problems

See the wiki page for more

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In Sympy is there a way to get a representation of Groebner basis in terms of defining polynomials?

In Sympy one can obtain the representation of a polynomial's reduction by a Grobner basis: from sympy import groebner, expand from sympy.abc import x, y f = 2*x**4 - x**2 + y**3 + y**2 G = groebner([x**3 - x, y**3 - y]) Q, r = G.reduce(f) assert f…
KludgeShySir
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Solving a system of cubic polynomials (to find intersection of Bezier curves)

Can anyone suggest a fix or an alternate route to find the solutions to this system? In particular I only care about solutions (s,t) in [0,1]x[0,1] Note: I'm looking for the intersection of two cubic Bezier curves here. I need the method to be…
mathandy
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How can you compute a simplified Groebner basis in GAP?

I tried to use Buchberger's Algorithm (see this or this) to compute a Groebner basis for an ideal in rational field. Here is my GAP script: F := Rationals; R := PolynomialRing( F, [ "x", "y", "z" ]); x := IndeterminatesOfPolynomialRing(R)[1]; y :=…
jscoot
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Groebner Base algorithm for Octave

I want to find a code or general algorithm to calculate the Groebner Basis of polynomials for Octave. I tried this: How do I compute a Groebner Base of a system of equations in Matlab but I get thr following error: error: strcat: inputs must be…
mrMath
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Using div() on multivariate polynomials in Sympy: incorrect remainder?

I want to compute remainders in Python for multivariate polynomials and I found that div() from sympy should do the trick (I also need sympy for Gröbner computations). But the problem I keep finding is that it seems that div() only checks the…
Robert
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Solving systems with Groebner basis

Suppose that a finite set of polynomials in C[x,y,z] has a finite number of solutions (i.e. the generated ideal is 0-dimensional). Suppose also that the Groebner basis with respect to lex order x>y>z is [f(z), g(y,z), h(y,z), k(x,y,z)] As well…
V. Vil
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Prove Theorem with Groebner Basis

I'm trying to prove some theorems using Groebner Basis (as described in Cox, Little and O'Shea Link ) The mentioned book gives as an excercise to prove Pappus theorem using the given methodology, but I really can't make it work. I've tried usign…
Francisco Martínez
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Unexpected effect of smt.arith.nl.gb on reasoning with (syntactic) equality - bug?

Consider the following SMTLIB program (on rise4fun here): (set-option :auto_config false) (set-option :smt.mbqi false) (set-option :smt.arith.nl.gb false) (declare-const n Int) (declare-const i Int) (declare-const r Int) (assert (= i n)) (assert…
Malte Schwerhoff
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Sagemath Singular interface crashing when computing Groebner basis

I am trying to compute the Groebner basis for a large number of polynomials using Sagemath and its Singular interface. However, for some (largest) polynomials, Singular is crashing on the HPC I'm running my code in. My code simply calls…
tomate
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Deriving Heron's formula with Sympy using Groebner basis

I'm trying to implement with Sympy the procedure described in this lecture, where Groebner basis is used to derive Heron's formula, but I'm not able to implement the last step, the computation of the correct Groebner basis, surely because of my lack…
mmj
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Measure and bound time spent in arithmetic sub-solvers

Q1: Is it possible to query the times Z3 spent in different sub-solvers? Calling (get-info :all-statistics) gives the overall run time of Z3, but I would like to break it down into individual sub-solvers. I am particularly interested in the time…
Malte Schwerhoff
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