Gröbner Bases: A Short Introduction for Systems Theorists
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Citations
Attacking Bivium using SAT solvers
Yukawa Couplings in Heterotic Compactification
Deriving Finite Sphere Packings
Appearance of multiple stable load flow solutions under power flow reversal conditions
References
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra
Gröbner Bases: A Computational Approach to Commutative Algebra
An Introduction to Gröbner Bases
Efficient Computation of Zero-dimensional Gröbner Bases by Change of Ordering
Related Papers (5)
Frequently Asked Questions (8)
Q2. What is the elimination property of Gröbner bases?
The elimination property of Gröbner bases guarantees that, in case G has only finitely many solutions, G contains a univariate polynomial in x .
Q3. What is the main theorem of Gröbner bases theory?
The main theorem of Gröbner bases theory then shows that, given a finite F , if you "master" the finitely many S polys, then you master the infinitely many polynomials that allow two or more essentially different reductions.
Q4. What is the importance of tuning the Gröbner base algorithm?
For the practical implementation of the Gröbner basis algorithm, tuning of the algorithm is also important, for example byheuristics and strategies for choosing favorable orderings of power products and for the sequence in which S polynomials should be selected etc,good implementation techniques and data structures.
Q5. What is the lexicographic order of the Gröbner base?
If the authors used the lexicographic order that ranks x higher than y then, correspondingly, the Gröbner basis would contain a univariate polynomial in y .)
Q6. What does Theory tell us about the vanishing polynomial?
Theory tells us that, whatever the resulting polynomials in y will be, they will always have a nontrivial greatest common divisor which, in fact, is just the non vanishing polynomial of lowest degree.
Q7. What was the first use of the Gröbner bases method?
the method of Gröbner bases was introduced in [3, 4] for the algorithmic solution of some of the fundamental problems in commutative algebra (polynomial ideal theory, algebraic geometry).
Q8. What other software systems are mainly based on Gröbner bases?
there exist special software systems that are mainly based on the Gröbner bases technique, for example, CoCoA [12], Macaulay [17], Singular [18].