Talk:Gröbner basis

message: If I'm not mistaken, this algorithm generalizes not only Euclid's algorithm but also Gaussian elimination and the simplex algorithm.  I don't know enough about it to incorporate those topics into the article.  Can someone do that? 128.101.13.104 02:15, 6 Nov 2003 (UTC)
thread_title: Untitled

message: The text implies that a polynomial system with finitely many solutions can be "solved" using the Gröbner basis. But if the system has zero solutions, then this technique won't always demonstrate that fact, right? Perhaps the article should say "If the system has a finite nonzero number of solutions..." or some such. I'd change it myself, but I'm a bit new to Gröbner bases, and would like some assurance that I'm right. Staecker 17:46, 25 Jun 2005 (UTC)
replies: text: In fact, if there is no solution, then by the Nullstellensatz the polynomial 1 belongs to the ideal, and since its leading monomial is 1 and is the smallest element in every term order, the reduced minimal GB will be exactly {1}, so You would know. For the other thing, look up lexikographic order, FGLM (Jean C. Faugere is the F), RUR by Fabrice Rouillier, also called "geometric solution" in other contexts. A first result on finite numbers of solutions is that for each variable there has to be a polynomial in the GB with a pure power of said variable as leading monomial.--LutzL 14:28, 29 August 2005 (UTC)
thread_title:  About polynomial systems 

message: Copied from User talk:LutzL for possible use by others. --LambiamTalk 14:49, 5 May 2006 (UTC)
replies: text: As I understand it now, the standard bases are defined for ideals in the ring of Puisseux series. Thus they contain Gröbner bases as a special case. The real contribution of Buchberger to the fundamentals is the proof that his algorithm stops in finite time. Perhaps it should be mentioned somewhere that the idea comes from the non-constructive proof by Hilbert of the Basissatz (Kronecker had a constructive proof, but only formulated for  bivariate polynomials) -- No, I'm not an expert in the whole of the topic of Gröbner bases. They are interesting to me in the aspect of their inefficiency. I got most of my knowledge from Gethgen/vonzur Gathen: Modern Algebra, where they mention H. Hironaka  in the second sentence of the introduction. And from M. Guisty: Bases standard, élimination et complexité, notes of a lecture given at X, where some propositions are attributed to Hironaka.--LutzL 07:29, 5 May 2006 (UTC)
thread_title:  Gröbner basis and Hironaka 

message: The definition of reduced bases didn't make sense.  As it was written, it referred to the "ideal generated by the leading coefficient", which is always all of R.  Therefore, the condition for a reduced basis could never be satisfied.
I changed "coefficient" to "term".  Now at least it at least makes sense, although I don't have a reference to be sure whether it is correct.  If anyone would care to look it up and check it that would be good.  When I find a reference I will do so.
Heatkernel 10:58, 8 May 2006 (UTC)
thread_title:  Definition of Reduced Gröbner Bases 

message: Solving equations is the elementary application which, from my point of view, motivates the theory. The formulas, { f1[x1, ..., xn] = a1, ..., fm[x1, ..., xn] = am} and g(xn) = b, can be simplified into {f1[x1, ..., xn] = 0, ..., fm[x1, ..., xn] = 0} and g(xn) = 0 . For example a circle cutting a straight line: x2+y2−4 = y−1 = 0. You do not need to understand ideals in order to understand the problem of solving systems of equations with several unknowns. Bo Jacoby 08:17, 8 August 2007 (UTC)
replies: text: Hi Bo, nice to see You again. And now, what was the point? The main application of GB, as it is my experience, is to prove structural theorems in algebraic geometry. Such as estimates on the Hilbert polynomial, number of rational points etc. Your point is valid, but the number of solvable systems is rather small.--LutzL 06:45, 9 August 2007 (UTC)
thread_title:  Solving equations 

message: An IP hinted at a paper "Les Invariants des formes binaires" in Jour. Des Mathematiques Pure and Applique'es, (1900), se'rie 5/ T 6, pp 141-156.", which is available from a digitisation project site (PDF). There exists an english translation at a Gröbner bibliography project (PDF) which claims it as an early example of a computed Gröbner basis. Perhaps someone might include in in the article.--LutzL (talk) 09:37, 6 December 2007 (UTC)
thread_title:  Earliest examples 

message: I wish this article had some examples.  I was not able to get a clue about what was going on even though I know what a polynomial ring is and I know what an ideal is.  —Mark Dominus (talk) 14:58, 12 October 2011 (UTC)
replies: text: Rather than add an example, I added an intuitive description. If you still think that is insufficient, please say so. Simplex (talk) 15:02, 21 March 2013 (UTC)

text: See above under "Example of Groebner basis"--Lieven Smits (talk) 21:09, 9 April 2014 (UTC)
thread_title:  Examples please 

message: Their article differs substantially from ours. We could improve ours by using information from theirs. Here is the Google Translation of http://ru.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BC%D0%BD%D0%BE%D0%B3%D0%BE%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%B8%D0%B5:
röbner basis of some ideal I of the algebra of polynomials K [x_1, \ ldots, x_n] concerning the order of " \ Prec "On monomials - is a finite set of polynomials G K [x_1, \ ldots, x_n] such that the highest (relative \ Prec ) Member of each polynomial of I divided by a senior member of at least one polynomial of G. The order \ Prec must be complete and in addition to the following two conditions:
    Multiplicativity: x ^ a \ prec x ^ b entails x ^ {a + c} \ prec x ^ {b + c} .
    Minimum unit: 1 \ prec x ^ a For a> 0 . 
Content
    1 Types of Gröbner
        1.1 Minimum Gröbner
        1.2 The reduced Gröbner 
    2 Algorithms for constructing
    3 Applications
    4 History
    5 Notes
    6 Links 
Types of Gröbner
Minimum Gröbner
Minimal Gröbner basis of a polynomial ideal I is called its Gröbner G, such that:
    Coefficient of the leading monomial of each p \ in G equal to one.
    Each leading monomial p \ in G is the leading monomial of any other q \ in \ {G \ setminus {p} \}
The reduced Gröbner
Reduced Gröbner basis of a polynomial ideal I is called its Gröbner G, such that:
    Coefficient of the leading monomial of each p \ in G equal to one.
    Each leading monomial p \ in G not divisible by the leading monomial one other elements of the basis. 
For the reduced Gröbner basis of the ideal we have the following statement:
Let I - polynomial ideal, and given a monomial ordering. Then there is a unique reduced Gröbner basis of the ideal I.
Algorithms for constructing
The earliest construction algorithm reduced Gröbner basis of the ideal is Buchberger algorithm . Interestingly, the well-known Gaussian elimination for solving systems of linear equations is a special case of the Buchberger algorithm.
In addition, the French mathematician Jean-Charles Fougeres proposed algorithms F4 and F5 finding Gröbner basis.
Applications
Gröbner basis is the most important concept of computer algebra , algebraic geometry and computational commutative algebra .
History
Austrian mathematician Wolfgang Graebner developed the theory of standard bases for the free commutative case in the early 30's, and published it in 1950 , [1] where he wrote:
'	
I started using this method 17 years ago for a variety of examples, including the very complex.
The original text (in German) [показать]
	'
In 1964 a similar concept for the local rings was developed Heysuki Hironaka, received Fildovskuyu Prize in 1970 . He introduced a system of polynomials called the standard basis.
The concept of Gröbner basis was introduced in 1965 an Austrian mathematician Bruno Buchberger , a former student of Graebner. Buchberger proposed a constructive procedure for constructing Gröbner basis in the form of efficient computer algorithm, which was later called Buchberger algorithm .
The existence of a standard basis of the ideal is based on the "lemma of composition", which was first proved for the most complex of the known cases (free Lie ) A. Shirshov . [2] The correct similar result for the free associative and commutative case was considered obvious and did not attract much attention until later works LI bokuto on the theory of attachment in the body of associative rings and rings with specified properties. In 1972 L. I. Bokut published "lemma Shirshov's Song" for the free associative case the notes of a course on associative algebras Novosibirsk University. From this and oral communication, it became known to American algebraist J. Bergman, who published it in 1979 under the name of «Diamond Lemma» (Diamond Lemma), however, strong evidence in missing and was identified only mimic diagram of "fusion" needed to understand the idea of ​​the Shirshov composition. After the publication of Bergman "Diamond Lemma" became popular among algebraists and geometers, it also drew attention to the "Gröbner basis" Buchberger. In the mid 80's the standard basis for superalgebras and color Lie was built Moscow algebraists AA Mikhalev.
Notes
    ↑ W. Gröbner (1950). " Über Die Eliminationstheorie ". Monatshefte für Mathematik 54: 71-78.
    ↑ CSF, 1962, Volume 3, № 2, p. 292-296. 
References
    Pankratiev EV Introduction to computer algebra .
    Arzhantsev IV Gröbner bases and algebraic equations . - M.: MCCME , 2003. - 68. - 
    Churkin VA Systems of polynomial equations, ideals and their bases, dividers .
    D. Cox, J. Little, D. O'Shea, Ideals, varieties and algorithms. Introduction to computational aspects of algebraic geometry and commutative algebra.
    Bernd Sturmfels. What is a Gröbner Basis? 
Crasshopper (talk) 16:23, 19 February 2013 (UTC)
thread_title:  Info from the Russian Wikipedia 

message: :The study of ideals is fundamental to commutative algebra and algebraic geometry.  Ideals contain infinitely many elements, so, as in linear algebra, it is preferable to obtain a finite generating set with certain desirable qualities.
:It is no surprise that not all generating sets possess these qualities; after all, in linear algebra one typically desires a basis of the vector space, which, in addition to generating the space, can contain only elements that are linearly independent. This property allows us to compute easily properties of the space, such as its dimension.
:In this vein, a Gröbner basis is a generating set of an ideal, with certain desirable qualities that allow one to answer important questions about the ideal or an object it describes. The similarities do not end there; as noted already, the computation of a Gröbner basis generalizes Gaussian elimination of a system of linear polynomials to non-linear polynomials.
</blockquote>
:As explained above, I had added it in lieu of an example. I wish you had taken the time to look through the discussion first, but if you find it unhelpful for the task, I shan't bother putting it back. Simplex (talk) 17:56, 18 March 2014 (UTC)
thread_title:  Intuitive notion 

message: Is this still up to date? The anwsers to this question on
Computational Science Stack Exchange and this benchmark seem to suggest that the fastest implmentation is now mgb, the Gröbner basis library shipped with Maple. 212.201.44.244 (talk) 18:36, 24 January 2018 (UTC)
thread_title:  Performance of Implementations