FROBBY USER MANUAL - VERSION 0.9.0 September 23, 2011
Contents
1. b c73 a 2 c 2 a 3 b Here the monomial generator bc corresponds to the irreducible ideal b c 2 7 Primary decomposition Frobby can compute primary decompositions as well as irreducible decompositions We need an example where there is a difference between these two things so type this into a file named input3 R QQId b cl I monomialIdeal dxb c b72 c b 10 d 10 To get the primary decomposition type frobby primdecom lt input3 which produces the decomposition R QQ d b c I monomialldeal d 10 b710 c I monomialIdeal d b d 10 b 2 Note that these two ideals are primary and that their intersection equals the original ideal you should know how to check the intersection yourself by now An irreducble decomposition is also a primary decomposition since an irreducible ideal is primary but the irreducible decomposition can have many more elements than the primary decomposition does In this case it has one more since we get the irreducible decomposition by typing frobby irrdecom lt input3 which produces the output QQld b c monomiallIdeal H wv oot 10 BJARKE HAMMERSHOLT ROUNE I monomiallIdeal d 10 b I monomiallIdeal The primary decomposition is not unique but Frobby computes the nicest pri mary decomposition which is the one gotten by intersecting irreducible components of the same support and that primary decomposition is uni2. ab By default Frobby will compute the Alexander dual of the input according to the point that is the least common multiple of the minimal generators of the input ideal To use some other point append that monomial at the end of the file We can append a line by using echo and cat like this 6 BJARKE HAMMERSHOLT ROUNE echo a 10 b710 c710 cat input which prints R QQla b cl I monomialldeal b c a 2 c 2 ax b a7 10 b710 c710 so if we feed this to alexdual like this echo a 10 b710 c710 cat input frobby alexdual we get the Alexander dual of the ideal according to a b c R QQ a b cl I monomialIdeal b710 c78 a79 c79 a 8 b710 which is To see the options that alexdual accepts type frobby help alexdual Also note that you generally don t have to type actions out you only have to specify a unique prefix So to get the Alexander dual it is sufficient to type frobby al lt input However it is not enough to just type frobby a lt input 66 Since this is ambigous with the actions assoprimes and analyze that also have a as a prefix In this case Frobby reports that ERROR Prefix a is ambigous Possibilities are alexdual assoprimes analyze 2 4 Hilbert Poincar Series To get the Hilbert Poincar series type frobby hilbert lt input which produces the output R QQla b cl p 1 b c 2 a 2 b a7 2 b c72 a73 c73 a 3 b c73 This means t
3. t type the file in correctly Note how transform produced output in the Macaulay 2 format If you want to transform your file into a different format you can do so by typing frobby transform lt input oformat monos which will produce output in a format that can be understood by the program Monos It will print vars a b C a 2 b b c 2 a 3 c73 l In general Frobby uses the first non space character of an input file to figure out what format it is in and it then produces output in that same format unless you specify the input or output format explicitly using the command line options oformat and iformat In general you tell frobby to do something by typing frobby ACTION OPTIONS where ACTION is something to do and OPTIONS is a possibly empty list of options that specify how to do that thing Thus you can make the transform action behave in a number of different ways depending on the options you give it To take a look at everything that transform can do type frobby help transform In general you can get information on any action by typing frobby help ACTION where ACTION is the name of the action To get a list of all actions type frobby help 2 3 Compute the Alexander dual of an ideal To compute the Alexander dual of our ideal type frobby alexdual lt input which produces the output R QQ a b c I monomialIdeal b c a 2 c 2 a b And it is indeed true that the Alexander dual of ab bc ac is be a c
4. install To see if your installation of Frobby is working type frobby You should now be looking at information about Frobby and its help system If you instead get a message along the lines of command not found type bin frobby If that works the install script was not able to place Frobby in a convenient place on your system probably because you don t have the access rights to install programs centrally on your system In that case you can simply call Frobby as bin frobby from the directory where you unpacked it or you can place the binary somewhere on your path where you do have access rights to put it If you want to check that your Frobby compiled correctly type make test This will run a comprehensive battery of tests on Frobby which can take a while especially on Cygwin which is generally a slower platform 2 TUTORIAL This is a tutorial on how to use the command line interface for Frobby FROBBY USER MANUAL VERSION 0 9 0 3 2 1 File formats All of the file formats that Frobby understands are files that will make sense to some other piece of mathematical software as well In general Frobby will figure out by itself which of the possible formats you are using and will produce output in that same format so this is not something you normally have to think about In this tutorial we will be using the Macaulay 2 format which looks like this R QQ a b c d I monomialIdeal a2 a b 5 c y This describe
5. 2 and put it into the empty 4ti2 subfolder of the Frobby folder you can use an algorithm that is much faster when the Frobenius number itself is moderately large Assume that you have installed 4ti2 in that folder Then within the Frobby folder type frobgrob frob which will produce the same output namely 29 Now put this input into a file named frob2 1234567890001 348461546433 6484646532513541 45464188888115164 1561484651561864468465310 and type frobgrob frob2 to get that the Frobenius number is 15111053020091472900 This computation would never have completed using the dynfrob action Note that you get better performance if you sort the numbers in increasing order Note that frobgrob is a script that uses 4ti2 and the frobgrob action of Frobby This is a bit annoying to get right which is why there is a script doing it To get a hint on how to do this yourself directly without using the script take a look at the script look at the documentation for 4ti2 and type frobby help frobgrob to get an idea about what input frobgrob expects You can also generate random Frobenius instances by typing frobby genfrob to produce something like 4259 8346 12295 18936 22645 27086 34182 69298 74617 84570 The output is random so you will likely get a different instance 2 12 Analyze an ideal To get Frobby to analyze the ideal and tell you what it figured out about it type frobby analyze lt input which produces the output 3 g
6. FROBBY USER MANUAL VERSION 0 9 0 BJARKE HAMMERSHOLT ROUNE September 23 2011 CONTENTS 1 Installation 1 1 1 Cygwin preparation 2 1 2 Mac OS 10 5 preparation using Fink 2 1 3 Linux or Mac preparation without Fink 2 1 4 Installation of Frobby 2 2 Tutorial 2 2 1 File formats 3 2 2 An example file and transformation 4 2 3 Compute the Alexander dual of an ideal 5 2 4 Hilbert Poincar Series 6 2 5 Intersection of ideals 7 2 6 Irreducible decomposition 8 2 7 Primary decomposition 9 2 8 Associated primes 10 2 9 Transform a polynomial 10 2 10 Maximal standard monomials 11 2 11 The Frobenius problem 11 2 12 Analyze an ideal 12 This is a minimal manual for Frobby At present it contains information on how to install Frobby on your system as well as documentation in the form of a tutorial which shows how to do most things that Frobby can do The most complete reference for the functionality of Frobby is the help system built into Frobby However that system may be too complete to serve as a good introduction and this manual is intended to cover that area This manual will be expanded in time to cover the algorithms that Frobby uses explain the file formats in detail and cover the mathematical meaning of the com putations that Frobby performs 1 INSTALLATION Frobby is tested to work on Linux Mac OS 10 5 and Cygwin on Windows It may work on other platforms but that has not been tested Although the procedure to ins
7. als type frobby ptransform lt pinput oformat 4ti2 to get 6 4 1 000 Sl aL 1 212 1 303 1 313 coefficient a b c 012 210 The first line indicates the size of the matrix and each row of the matrix is a term with the first number being the coefficient and the remaining numbers being the exponent vector I e the exponent of a b and c respectively in that order To see what else ptransform can do type frobby help ptransform Note that if we knew in advance that we wanted to change the format to say 4ti2 format we could have produced the file pinput in that format directly by typing frobby hilbert lt input gt pinput oformat 4ti2 2 10 Maximal standard monomials To produce the maximal standard mono mials type frobby maxstandard lt input3 to get R QQ d b c I monomialIdeal d 9 b 9 2 11 The Frobenius problem Given an input vector p p1 Pn of positive relatively prime integers the Frobenius number of p is the largest integer that cannot be written as a linear combination of p1 Pn where the coefficients in the combination are non negative integers To work with this put the following example in a file named frob 6 10 15 and type 12 BJARKE HAMMERSHOLT ROUNE frobby frobdyn lt frob to get as output that the Frobenius number is 29 The frobdyn action uses an obscenely slow dynamic programming algorithm to compute the Frobenius number If you install the program 4ti
8. enerators 3 variables This is admittedly not an impressive level of analysis being performed To see a few more things that analyze can do type frobby help analyze FROBBY USER MANUAL VERSION 0 9 0 13 The idea is that this action will be expanded to provide actual valuable information such as whether the ideal is strongly generic if it is weakly generic strongly weakly co generic how many irreducible and primary components it has if any of the given generators are non minimal and various things like that
9. hat the numerator of the Hilbert Poincar series of ab bc ac is 1 be ab a2bc a abc FROBBY USER MANUAL VERSION 0 9 0 7 so the series itself is 1 bc a b abc abc abc 1 a 1 b 1 c This is the multivariate N graded hilbert series You may want the numerator of the univariate Hilbert Poincar series which you get by typing frobby hilbert lt input univariate This produces the output R QQ t p EOT t 6 t75 2 t 3 1 so the univariate Hilbert Poincar series is t t 2841 Frobby always uses the variable t for the univariate series The univariate series is gotten from the multivariate series by substituting t for each of the variables in the ring in this case a t b t and c t 2 5 Intersection of ideals We will now intersect two ideals so we need another file with an ideal in it Type the following ideal into a file named input2 R QQ d b c I monomialIdeal dxb c b72 c This is the ideal dbc b c To intersect that with the ideal a b bc a3c we will need to concatenate the two files representing these two ideals This is done by typing cat input input2 which produces the output R QQ a b cl I monomialIdeal a 2 b b c 2 a 3 c 3 R QQ d b cl I monomialIdeal dxb c b72 c If we pass this to the intersect action we will get the intersection So type cat input input2 frobby inte
10. of a and b and a single indeterminate named ab Thus on this input R QQ a b cl I monomialIdeal a 2b b c 2 c 3 a 3 Frobby will produce the following error SYNTAX ERROR format m2 line 3 Expected but got b Frobby is expecting a right parenthesis because that would make sense after the 2 on line 3 seeing as there is no there Note that Frobby does not care about additional space and it does not distin guish between a space and a newline To see a list of all file formats that Frobby supports type frobby help io To see an example of a monomial ideal in a format X type frobby genideal oformat X If you do not specify a format the Macaulay 2 format will be used so if you type simply frobby genideal You will get something like R QQ x1 x2 x3 I monomialIdeal x1 73 x275 x376 x17 2 x276 x3 x175 x272 x378 x1 78 x2 x373 X175 x274 x377 However genideal generates a random ideal so every time you run it you will likely get a different ideal 2 2 An example file and transformation We will need an example to run the following commands on so take the following lines and put them into a file called input R QQ a b cl I monomialldeal a 2 b b c 2 c 3 a 3 J FROBBY USER MANUAL VERSION 0 9 0 5 To check that you wrote the format correctly try to get Frobby to read the file by typing frobby transform lt input If you get an error from Frobby then you didn
11. que 2 8 Associated primes You get the associated primes by typing frobby assoprimes lt input3 R QQ d b cl I monomialIdeal d b d b c Each generator encodes an associated prime In this case the associated primes are d b and d b c You can check that these are the radicals of the primary components 2 9 Transform a polynomial Transform does a number of things to ideals and ptransform is the corresponding action for polynomials Put a polynomial into the file pinput by typing frobby hilbert lt input gt pinput Then type cat pinput to get R QQ a b cl p 1 b c 2 a 2 b a 2 b c72 a 3 c 3 a7 3 b c73 This is a file that Macaulay 2 can understand To change the format of this file to something that Singular can understand type frobby ptransform lt pinput oformat singular to get ring R 0 a b c 1p int noVars 0 poly p 1 FROBBY USER MANUAL VERSION 0 9 0 11 b c 2 a 2 b a 2 b c 2 a 3 c 3 a 3 b c 3 You might notice that the Singular format has something called noVars This is there merely because all Frobby formats must be able to represent a ring with no variables and Singular has no such concept Thus Frobby gets around that by always giving the ring at least one variable but writing noVars 1 if there really should not be any and noVars 0 otherwise To get the polynomial into a format that 4ti2 might have produced if it had a concept of polynomi
12. rsect 8 BJARKE HAMMERSHOLT ROUNE to get the output R QQ a b c d I monomialIdeal b c 2 d b 2 c 2 a 2 xb c d a7 2 b72 c Note that we now have 4 variables in the first line since the two ideals together used these 4 variables We see that the intersection is bed b c abcd a b c 2 6 Irreducible decomposition To get the irreducible decomposition type frobby irrdecom lt input which produces the output R QQla b c I monomialldeal gt c73 I monomialIdeal a 2 c 2 I monomialldeal a3 b Note how the list of the variables does not have to be repeated because all three ideals lie in the same ring This is saying that the irreducible decomposition of ab be a 0 is a c a Then it should be true that the intersection of these three ideals is the original ideal and indeed if we type frobby irrdecom lt input frobby intersection we get the output R QQla b cl I monomialldeal b c 2 a 2 b a 3 c 3 23 This format for a list of the irreducible components takes up a lot of space and is not the easiest to read We can get a more compact notation by encoding each irreducible ideal as the product of its generators noting that this is a bijection between irreducible ideals and monomials Do this by typing FROBBY USER MANUAL VERSION 0 9 0 9 frobby irrdecom lt input encode to get the output R QQ a b cl I monomiallIdeal
13. s the monomial ideal a axb c as an ideal in the polynomial ring Qa b c There is some syntaxtic noice in this file format such as I mono mialldeal which is solely there so that Macaulay 2 will understand the file In general if the variables are z1 n and the generators are g1 g then the corresponding file will look like R QQ x1 xn I monomialldeal gi gk It is important to note that Frobby does not understand Macaulay 2 code at all it merely reads this very specific file format This is why Frobby is so fussy about this precise syntax being used E g this is not valid T QQla b c dl I monomialldeal a2 a b 5 c The error is that it must be R and not T so Frobby will display the error SYNTAX ERROR format m2 line 1 Expected R but got T Likewise this will also result in an error R QQla b c d I monomialideal a 2 a b 5 Cc The error is that the second i in monomialldeal is lower case and it is supposed to be upper case Frobby will diplay the error SYNTAX ERROR format m2 line 2 Expected monomialIdeal but got monomialideal 4 BJARKE HAMMERSHOLT ROUNE There is one particular error that is easy to make namely that indeterminates in the same monomial must be separated by an asterisk This is because indetermi nates can have names of more than one character so otherwise there is no way to distinguish between the product
14. tall Frobby is the same on these platforms Frobby requires the GMP library which is installed in different ways depending on platform 1 2 BJARKE HAMMERSHOLT ROUNE 1 1 Cygwin preparation Before installing Frobby first run the Cygwin file setup exe and then install the following Cygwin packages libgmp devel In category libs For infinite precision integers make In category devel To build Frobby gcc g In category devel To compile Frobby diffutils In category utils For Frobby s test system which uses diff 1 2 Mac OS 10 5 preparation using Fink If you have fink installed on your system simply type fink install gmp shlibs libgmpxx shlibs 1 3 Linux or Mac preparation without Fink Download the newest version of GMP from http gmplib org and unpack it somewhere Then start a terminal go to the directory where you unpacked it and type configure enable cxx make make check make install Typing make check is optional but recommended to make sure that your sys tem compiled GMP properly If you have trouble installing GMP first consult the GMP manual and then ask for help on the gmp discuss gmplib org mailing list 1 4 Installation of Frobby This step is the same on all platforms First down load the latest version of Frobby from http www broune com frobby Then open a terminal go to the directory where you downloaded Frobby and type tar xvf frobby_v0 8 2 tar gz cd frobby_v0 8 2 make make
Download Pdf Manuals
Related Search