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# Commutative Algebra - Science topic

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other methods apart from commutative algebra, minimum number of colors, relationship between ideals with edges and nodes,
e.g., see
for a
"Timetable scheduling problem"
"mixed-integer (linear) programming model"
which is "usually" solved by a genetic algorithm method
Good Luck
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The unexploited unification of general relativity and quantum physics is a painstaking issue. Is it feasible to build a nonempty set, with a binary operation defined on it, that encompasses both the theories as subsets, making it possible to join together two of their most dissimilar marks, i.e., the commutativity detectable in our macroscopic relativistic world and the non-commutativity detectable in the quantum, microscopic world? Could the gravitational field be the physical counterpart able to throw a bridge between relativity and quantum mechanics? It is feasible that gravity stands for an operator able to reduce the countless orthonormal bases required by quantum mechanics to just one, i.e., the relativistic basis of an observer located in a single cosmic area?
What do you think?
A charged particle whizes past a very massive body, therefore its trajectory bends somehow
and radiates (for Classical reasons). Therefore there might be some radiation surrounding a
Black hole. So who is responsible? Not QM. Perhaps only Einstein or Newton?
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• We know ,euclidean algorthm is feasible in the set of integers.
• Taking motivation from this,we define an Euclidean Domain(E.D.) as follows:
R is an E.D. if it is a domain,andwe have a map d:R->non-zero integers,
such that, for given a,b(non-zero),
there exists q,r with
a=bq+r and d(q)<d(r).
• Now,this d function ,for any abstract ED,is the counterpart of |.| function,in case of intgers,
we have ,a=bq+r with,0 \le r < |b|.
• Now the question is:
[] It turns out (q,r) that exists for a given (a,b) in case of intgers ,is unique .
[A simple proof would be:if another (q',r') exists for same (a,b),
if q'=q,we have r'=r,and we are done.
and if,q' and q are distinct,bq+r=bq'+r' implies r' and r differ by multiplies of b,thus if 0 \le r < |b| holds,it is clear it won't hold for r',and thus we can never have (q',r') and (q,r) distinct,and we are done.]
[] But,one now would ask,is it true as well that for given (a,b) in any E.D. the existing (q,r) would be unique ? If yes,we need a proof,and clearly,the same proof does not work,as d(.) is much more generalized than |.| .Or,we need a counter eg!
__________________________________________________________________________________
Let K={|z|<=1} be the closed unit circle (a connected compact) in C, R=O(K) be the ring of holomorphic functions f on K with d(f)=#{zeroes of f in K}, R is a ED.
Let a=a(z)=z^2+2z+1, b=b(z)=z (hence d(a)=2, d(b)=1). Let h=h(z)=sin(z)/z, h(0)=1.
Then
1) z^2+2z+1=(z+2)z+1, q=z+2, r=1 (hence d(r)=0)
2) z^2+2z+1=(z+2+h)z+1-zh, q'=z+2+h, r'=1-zh=1-sin(z) (hence d(r')=0,
since sin(z)=1 iff z=\pi/2 +2\pi*k, k\in Z).
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In the definition of a group, several authors include the Closure Axiom but several others drop it. What is the real picture? Does the Closure Axiom still have importance once it is given that 'o' is a binary operation on the set G?
I think no need to include closure property, if it is given that 'o' is a binary operation on the set G.
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it is not clear for me
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What are the current trends in Commutative Algebra and its interactions with Algebraic Geometry? What are the important articles that one should read for starting to research in this area?
For Cryptographic purposes.
I hope you find the attached article is helpful.
Best regards
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I'm currently studying in mathematics major, undergraduate program and doing my very first scientific writing and research. The topic that I get is about "Generalized power series rings", but I still have have no firm understanding about it. I've tried to look some journals in research gate and save some of it.
Could you recommend me books or scientific journals for me to read so that I could have more in-depth understanding on the topic?
Any recommendations is highly appreciated and welcome,Thank you.
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What are the conditions for a commutative ring with unit R to verify the proprety: every locally cyclic R-module is cyclic ?
I recall that an R-module is cyclic if it is generated by one element.
And is locally cyclic if every localization by a prime ideal will be generated by one element as a R_p-module.
This is obvious when R is local, but whzt are the general conditions ?
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I am trying to understand automorphism group of zerodivisor graph. If R is an infinite Noetherian ring, The automorphism group of corresponding zero-divisor graph is infinite. Do there exist any infinite non-Noetherian ring such that the automorphism group of corresponding zero-divisor graph is finite.
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To make two square matrixs A and B satisfy the commutative law of multiplication, say, AB=BA, at least one of the following conditions is required:
Condition 1: A=B.
Condition 2: A=kI or B=kI. I refers to the Identity matrix.
Condition 3: Both A and B are diagonal matrices.
Any other conditions?
Dear professor
Paul Marcoux
,
In the set M_n of nxn square matrices, the Zero matrix commutes with all other matrices, and in particular Zero matrix commutes with any non- diagonalizable matrix. Obviously, commute does not imply diagonalizable.
If we consider the set D_n of the square diagonalizable matrices,
the statement: Commutative matrices are simultaneously diagonalizable
is true over the set D_n. Mathematically, this is what I exactly meant, because the other case is trivial and well-known.
So, your suggestion is to make a general statement:
Diagonalizable commute matrices are simultaneously diagonalizable.
This will exclude all trivial cases. :-)
Best wishes
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hi
thank you
@ Yasir,
(3)
Best regards
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Starting in polynomial extents, computational commutative algebra, branched into applications via Gröbner basis theory and the generalized concept of approximate commutative algebra. But what is next? Do you believe the computation extends to non-commutative algebra?
Some questions can be solved in a non-commutative graded case:
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I have that R is the k-algebra (k is a field) finitely generated by S={f1,...,fm}⊂k[x1,⋯,xn] and this set of polynomials is minimal with respect to inclusion (i.e., e do not have redundant elements). However, I know that the fi's are algebraically dependent. Can be the number m=|S| unique? What are the conditions?
Generally, this number is not unique.  For example, {x} and {x2,x+x2} are both minimal sets of generators of k[x]. For the subalgebra k of k[x1,...,xn] every minimal set of generators has 0 elements (duh). I suspect that this is the only example where the number of elements in a minimal set of generators is unique.
The situation is different in the graded case. The polynomial ring k[x1,...,xn] has a grading. If R is a graded sub-algebra then every minimal set of homogeneous generators of R has d elements, where d is the dimension of the k-vector space m/m2 and m is the ideal generated by all homogeneous polynomials in R of positive degree. If R is not finitely generated then d is infinite. Every minimal set of (not necessarily homogeneous) generators of R must have at least d elements.
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Let R be a commutative ring with unity. Given two algebras A and A' over a field K, both with R-grading form. Prove that B:= A \otimes_R A' (the tensor product of A and A') has a R²-grading.
Any tip of how to prove this?
Dear Kelvin,
I think you meant: B=A\otimes _K A' (i.e. tensor product over the field K). Next, we should prove this by definition: if a\in A and deg(a)=r\in R, a'\in A' and deg(a')=r'\in R are two R-homogeneous elements then we assign deg(a\otimes a')=(r,r')\in R2. It remains to show that this grading  is compatible with multiplication in B=A\otimes _K A' (which is not hard).
Best regards, Yuri
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Suppose there is a polynomial P(x) of degree ≤ n (and in characteristic zero). Then, if P(x) takes value 0 for n+1 different values of x, then all its coefficients are actually zeros.
Let, however, all these coefficients be integer, and each of them be not greater than 10 in absolute value. Then, one single condition P(11)=0 is clearly enough to guarantee that P(x) is identical zero.
Then it may be asked about similar results for an algebraic function of several variables, instead of our simple P(x). Most likely, there is already a big theory about such things, so why not learn it. The question is where I can find the best exposition of such theory.
"if P(x) takes value 0 for n+1 different values of x",  "then all its coefficients are actually zeros."
1) Then p(x) is of degree >=n+1
2) No reason , that all coefficients then are zero.
3) A polynomial can has zeros in some domain without be identically zero.  Please reformulate your additional information in the question.
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Let (R,m) be a Cohen-Macaulay local ring, with canonical module ω, and let I be an ideal of R.
Question:  Is the minimal number of generators of $Hom_R(ω, Iω)$  equal to the minimal number of generators of I?
The answer is affirmative  if R is Gorenstein.
Thank you so much.
Thank you so much, Ahmed Munir.
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Let J be an ideal in a polynomial ring R[x] and let p(x) be a monic non-constant polynomial, where R is a commutative Noetherian ring with identity.  Let J(p) denote the ideal of R[x] generated by the polynomials f(p(x)), where f ranges over the elements of J.  Clearly J(p) cannot require more generators than J.  Could J(p) require fewer generators than J?  I attached a partial attempt at a proof.
I did ask this question in mathstackexchange recently, but this is my first question on ResearchGate.
The condition you mentioned in your first paragraph is not the minimality I'm talking about, if I understood you correctly.  The one I'm talking about is that there is no set with fewer elements that generates the ideal.  In fact the minimality you mentioned (if I understood it) is never satisfied:  For example, if we have an ideal J = (f, g), then we have for example (g + 1)f + (-f)g = f.  For example in Z[x], J = (2, x) is minimally generated by 2 and x even though we can get both 2 and x as polynomial linear combinations of 2 and x.
Regarding your second paragraph, not every polynomial in J(p) is of the form g(p(x)) for some g(x) in J.  For example, xg(p(x)) is probably not of that form.  For example, if J = (x2) and p(x) = x3, then J(p) = (x6).  In this example, x7 is in J(p) but it is not of the form g(x3) for any g in J (or any g at all).
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see p. 333 of the attached paper.
Yes, you can use the same definition for a non-commutative associative ring R. Note, however, that the resulting Nagata extension will typically be a non-associative ring!
For arbitrary (a,m), (b,n), (c,p) the equation ((a,m)*(b,n))*(c,p)=(a,m)*((b,n)*(c,p)) is equvalent to
n\sigma(a)c + mbc + p\sigma(ab) = nc\sigma(a) + mbc + p\sigma(ba).
If R is commutative, then the above equation is obviously satisfied, but if R is non-commutative then we can not immediately tell.
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that is :
How to find formula formula for steady temperatures in thin semi-infinite plate y > 0 whose faces are insulated and whose edge  y = 0  is kept at temperature zero except for the partion  -1<x<1,  y=0  where it is kept at temperature unity. ?
Dear colleague Kultchitsky,
Problem of the solving the Dirichlet's problem for the upper semi-plane is very interesting and usefull. I asked the same question, and generally gave the answer, missing only some details at the end, (but if it is a problem they can be added)..
Using well-known invariance of Laplace differential equation in regard to conformal mappings, we exceed the Laplace's equation                                                                 Txx(x.y) + Tyy(x.y) = 0                                                                   to the Laplace's equations                                                                                                                Fuu + Fvv = 0,
can be found on the web-site
How to solve the Dirichlet's problem for the upper semi-plane ? - ResearchGate. Available from: https://www.researchgate.net/post/How_to_solve_the_Dirichlets_problem_for_the_upper_semi-plane [accessed May 18, 2016].
All best
Mirjana
wit
(1) +(2) becomes the following Dirichlet's  problem
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We have “big” and “small” in our science and the related numerical cognizing way to universal things around us （one of the mathematical cognizing ways）. So, we have “very big (such as 10001000)” and “very small (such as 1/10001000)”, “extremely big” and “extremely small”… in mathematics. When “infinite” came into our science and mathematics, we naturally and logically have “infinite big (infinities)” and “infinite small (infinitesimals)”.
Many people think we really have had many different mathematical definitions (given by Cantor) for infinite: infinity (infinite big) in set theory. So, when talk about the mathematical definition for “infinite”, people only think about “infinite big” but negate “infinite small”.
Should the mathematical definition for “infinite” cover both “infinite big (infinities)” and “infinite small (infinitesimals)” or only for the half: infinite big (infinities)?
Dear Geng,
From a formal point of view, the construction of what you call ‘infinitely big’ and ‘infinitely small’ is different.
Cantor's infinite cardinal hierarchy arises from his attempt to generalise the idea of equinumerosity from the finite to the infinite. The limit construction in mathematics comes from the need to define in a mathematically concise manner a concept that was used in physical theorising (17th-18th  century Wallis, Leibniz, Mercator, Kepler, ) where 1/\infty, dx, partial derivatives and tendencies of series were used without proper definitions. (Note that the limit concept handled both things ‘series diverging to infinity’ and a proper use of the ‘infinitessimal dx’ in derivatives.) Finally the construction of hyperreals which provides another meaning to the notion of an infinitesimal, only arose in the context of first order logics and non-standard models of field theory.
One should also note the difference between the context in which a concept was proposed, and what it was later argued to represent. (Eg. hyperreals were not introduced to represent infinitesimals, although they are now thought to allow such representation and maybe much better than the limit construction. Similarly Cantor’s hierarchy of cardinalities was not introduced to represent the ‘infinitely big’, but was a side product of the attempt to formalise equinumerosity in purely set-theoretic terms).
There is, in principle, no good or bad in ‘infinities’ as there is no good or bad in any definition, as long as it is correctly and well defined. The question is rather, what you need them for (pragmatics of mathematics), what they allow you to understand (epistemology of mathematics) and whether you can do without it (ontology of mathematics). Your normative question of what should or should not be defined will have other meanings from these different perspectives and different answers, depending on what kind of stance you take on pragmatics, epistemology and/or ontology.
I hope this helps,
Best,
Eric
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Hello,
Is it interresting mathematically speaking ?
Could we desambiguous more economically the grouping of moves notation by another operation than addition ? Is it inspiring to think about a pattern symbolic  one ?
I could not open your file. It says that it is corrupt. Can you please just give us one example or two of your suggestions?
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I need examples of
(1) Two finitely generated ideals in a ring R such that their intersection is not finitely generated and
(2) An element x in a ring R such that (0:x) is not finitely generated.
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Let P=k[x1...xn] be a poly over a field. Suppose that R=k[x1...xn]/I. The canonical module of R is $ω_R=Ext^{n−dim(R)}_P(R,P)$.
The question is that is there any upper bound for the min num of generators of ωR in terms of r=the min num of gen of I? and\or the embedding dim of R?
One may add more reasonable conditions to R, e.g. assume that R is Cohen-Macaulay?
and\or
Assume that I is almost complete intersection i.e. r=codim(I)+1.
A desired upper bound would be (r \choose 2) i.e. μ(ω_R)≤(r\choose 2).
Recall that min num gen ωR denoted by  μ(ωR) is also called the type of the ring R denoted by r(R). And the latter is one iff R is Gorenstein , this is a result of P.Roberts.
Let's also remind that in the CM case the question is tantamount to ask an upper bound for the last betti number in the min free resolution of R over P.
If I understand your question correctly, I think the answer is no.  For instance,
almost complete intersections I (which define C-M quotients) are linked to Gorenstein
ideals J, and the canonical module of R/I is J/(regular seq). But Gorensteins can
have arbitrarily large numbers of generators. For explicit examples, try codim 3 Pfaffians.
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Considering G a group and H its subgroup. <br />F a field, and F(G) a group ring. Since F(G) is a ring, we can consider also that F(H) is a subring, and then F(G) as a left or right F(H)-module.<br /><br /><strong>*</strong>I thought about writting an isomorphism between F(G)/F(H) and G/H. If one is finitely generated, I could say rank[F(G)/F(H)] = rank[G/H] = [G : H].<br />I am not sure about this proposal.
Hello, Kelvin,
consider the (right) transversal T for H in G, i.e. G=disjoint union_{t \in T} H.t  of left cosets, |T|=|G:H|. It is known that FG=direct sum_{t \in T} FH.t, one of the summands being simply FH (for, say, t=1). Each summand is a free FH-module isomorphic to FH. If you take the quotient FG/FH you get direct sum_{t \in T, t \ne 1} FH.t of free FH-modules of the cardinality (if finite) |T|-1=|G:H|-1. For instance, consider the case G=H, the quotient FG/FG is trivially 0. Best wishes, Yuri
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Knowing that R is a commutative ring and R[X] is a free R-module.
It seems obvious because any R-module can be written as quotient of a free R-module and any of its submodules.
As R[X²] is a submodule of R[X] over R, then it is an ideal of R.
It turn to simply show a bijective homomorphism, but I encounter problems describing the elements in this relations.
Hi, if I correctly understood the problem,
you have a direct sum R[x] =R[x^2]+x*R[x^2] of R-modules, second summand consisting of sums of odd powers of x. Both summands are obviously isomorphic, and both are isomorphic to R[x] as R-modules (substitute t for x^2). The quotient R[x]/R[x^2] is isomorphic as R-module to
x*R[x^2], hence to R[x^2]  and consequently to R[x]. Best regards, Yuri
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Let A be a commutative nil-algebra of nilindex n and dimension finite over an algebraically closed field of characteristic not 2, such that there exist a non zero ideal I such that I=AI. (but I \cdot I \neq I. because in this case, Dr Misha constructed a counterexample)
Then Null(A) \cap A^2 = {0}.
By A a commutative finite-dimensional algebra means a vector space of dimension finite over a field k with a generic simetric bilinear operator. In this particular case, additionally the product satisfy for every x \in A x^2 x^2= x^4. (Commutative power-associative algebras).
By nil-algebra of nilindex n means that for every a \in A, and every arrange of n products of a, this product is zero.
This particular kind of algebras are not associative, and  don' t have 1.
By Null(A) means the set of elements a \in A such that the right multiplication map R_a (or equivalent, the left map L_a) is the zero map.
Thanks everybody, and I'm sorry for all my mistakes.
Also I've found: Whether or not the new definition of simplicity is an actual restriction
is not known. That is, whether or not commutative powerassociative
nilalgebras without proper ideals exist (except of course
for the zero algebra of dimension 1) is still an open problem.
So it's a very serious problem (If it haven't  been solved yet.)
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I'm searching for a counterexample for these sentence " every nonessential ideal in C(X) is not necessarily a z-ideal"
Dear Enas,
I hhave cheeked.
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This is a part of an equivalent statements theorem in Gillman and Jerison, Rings of continuous functions.
May anybody help me in proving it?
Thank you Peter. What is the correct statement?
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I am trying to find an example of IF ring R (every injective R-module is flat) which is not von Neumann regular (every R-module is flat).
R = Z_4 is IF ring which is not von Neumann regular
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Has anyone actually compared them on a specific problem?
One should also add CoCoA (see lin below) to the list of the freely available computer algebra systems specifically dedicated to commutative algebra (and non-free one could also add Magma). As written in the thread mentioned by Artur, there is not real point in asking which of the three is the "best" one, as this depends where much on what is important for you: breadth of the library, speed of computation, ease of programming. The next question is whether you have some specific problem in mind (for which perhaps only one of the three has some code available) or whether you are asking generally. But even for a very specific problem there is usually no simple answer. We developed recently a new algorithm for computing resolutions and implemented it in CoCoa. Then we run a lot of benchmarks comparing it with Singular and Macaulay2: each system had some examples where it was faster than the other one. Computational commutative algebra is a very complex field and there is usually not a "best" solution to a particular class of problems.
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Suppose I have an affine algebraic variety, and I suspect that a certain element of its coordinate ring can be represented as a product of two elements having some nice form. Are there computer algorithms for such factoring?
I have now made some progress; the program the helped me was Singular.
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Let there be a ring r of polynomials in six indeterminates t,u,v,w,x,y with complex numbers as coefficients. Take then a quadratic extension (if this is the right word) R of r by adding a new letter z which is the square root of a polynomial p in our six indeterminates (these latter are, of course, transcendental over C). Polynomial p is homogeneous of 14th degree, if this matters. The question is: where can I read about (algorithms of) factorizing elements in R? Answer for only homogeneous elements will suffice (assuming z has degree 7), but if a factorization is not unique, I want all of them!
Remark. Straightforward attempts using primary decomposition algorithm in Singular proved to be beyond the capabilities of my computer.
Here is an article that discusses the type of factorization you mention. (For what it's worth, I regard this as a very good article in this particular area.)
On Factorization of Multivariate Polynomials over Algebraic Number and Function Fields.
Proceedings of ISSAC '09, 199–206, ACM Press, 2009.
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Is there some method to solve something like [A,B]=C, where B is the unknown operator?
Basically I want to solve the equation
[D,A]=iaLz,
where D is the Laplacian (in cylindrical coordinates), a is some constant, and A is the unkown operator.
It seems to me that a crucial point about your problem is that you don't just want to realize a given operator as a commutator, but as the commutator with a fixed (given) operator. While I don't know the answer to your question, you could try to attack it by studying the action on eigenfunctions for D. If [D,A]=iaLz, then for any eigenfunction f of D, you get a differential equation on Af . Maybe this helps in solving the problem or spotting problems.
I see another possible approach, provided that one has a good understanding of [D,Lz] and of iterated commutators of this form. If this is the case, one could try to understand the Lie algebra generated by D and Lz .
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I know this question has answer for when G is infinite cyclic group group.Does there is a general proof?
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Hi all
I add here a pdf file about my question. The question involves an "easier" parametrization which I found and is described here. This is not a matter of  life and
death, but if somebody can think something easier I would be very grateful.
I can live with the parametrization that I found, but what is bugging me, is the fact that
I might be missing some much easier and obvious parametrization...
kind regards
Samuli Piipponen
Have a look at I. R. Shafarevich, Algebra I, paragraph 1, 2,3,4 at least, where there are many interesting examples of "coordinatisations".
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Type means the vector space dim of the first non zero Ext w.r.t R/m
Thanks Olgur. But does still such an emaple exist if we suppose that R is unmixed of dimension 2?(Unmixed = every associated prime is of height zero)
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I know the Milnor and Tjurina numbers reflect in certain way the complexity of
a singular point for a variety defined by one polynomial V=v(<f(x_1,...,x_n>)).
Now I know there exists some generalizations (or at least attempts have been made to generalize) similar quantities for a variety defined by more than one polynomial V(f_1,...,f_n).
For example the Milnor number should generalize to object called 'Milnor Class' in Chow's group.
I have mainly 2 questions regarding these possible generalizations:
1) Does anybody know a good reference book in which such problems would have been investigated, discussed and possibly rigorously defined.
2) Are there any known algorithms to compute such objects ? And if so does anyone know an article or a book where such algorithms could have been described ? Or at least partial results ?
Hi Rogier
Thank you for very much for your answer and book references are always appreciated! Yes I know these are implemented somehow in Singular. In fact Singular is one the programs that I mainly use. Unfortunately the cases we are dealing with are not always complete intersections, although in most cases they are and are generated by codim number of generators.
Also are you aware about the Hilbert-Samuel function/multiplicity number for singularities ? Do you remember any good books where the algorithms and def.s for these would be introduced in . I would really appreciate all book tips.
Also about a question about the 'intersection multiplicity': If two varieties intersect
transversally do you remember if there are kind of 'straight' or sequence of commands to verify this with Sage, Singular, Macaulay2 or CoCoa without coding the algorithms completely yourself ? Just to save time.
Also if you remember good books where these problems are tackled it
would also be good to know !
But thanks again for you response !
regards
Samuli Piipponen
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Is there a connection to computer science (theoretical), statistics, etc? I know algebraic statistics is a field where they study graphical models by viewing it as algebraic varieties, but I don't know much. Does anyone know of any applications?
Hi Tribid
Commutative algebra has various applications outside pure mathematics. You probably should look more computationally oriented books, where some chapters and exercises deals especially with applications. Look for example the two books of,
auths. Cox, Little, & O'Shea
1. Ideals, Varieties and algorithms
2. Using commutative algebra
Besides the the theory the cover a range of various applications.
You can also look at my publications list if you are interested :
seems that only few is -).
regards
- Samuli
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