Gröbner bases over K<X> and Z<X> in theory and practice

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Gröbner bases over K<X> and Z<X> in theory and practice
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We report on the recent theoretical adoption and implementation of non-commutative Gröbner bases both over fields and over the ring Z with more principal ideal rings in mind, which was accomplished in parallel. In both cases we are based on the Letterplace correspondence of La Scala and Levandovskyy. The corresponding subsystem of Singular is called Letterplace. In addition to division with remainder algorithm, we also provide algorithms for syzygy bimodules and lifting. Despite a certain similarity, the case of rings as coefficients demonstrates crucial intrinsic differences, compared to the case of fields. We also remark on models of computation and on decidability.
Keywords Letterplace Singular Gröbner basis Gröbner bases

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the play's productions little wine industry is not stuff and i where can you do this for a nation we are the authors and we give you talk about global basis over heels and rings in theory and practice at the icy in this conference.
relate to or topic there were two players recently published the ice conference and now i will begin with short introduction to what that the place isn't what it does so it is a subsystem of the computer under a system single and next to plural and next this year that the days.
is used for finally presented associate that address all the effect the fields including also free eyebrows. and since recent is also possible the to do these computational effect of rings such us into just and things have the support of local in a nation or brings you can also computer or work cautions of this these rings.
the name comes from a correspondence between the free associate of a grand scale x. and a commuter the euro which is infinitely generated were on which we know by k x am still the elements the in the tone is of this community as well. i'm very well as exciting with i'll referring to the letter n.j. referring to place and the ideals in this infinite generated commuter the euro i called the letter place ideas so now if you can compute the global basis. but let's replace ideal then is possible to transfer the spec to look no basis in the free associate with this theory will serve you well a play about the skyline weekly menosky since two thousand and nine.
and in the modern implementation of thing let place you can now compute. the any objects in this the industry associated i'd run but internally using this that this coins correspondence are low computationally done with the commuter the day the structure. the gloom of technology is certain to fold we once earth and the glittering team which shown consists of one being able to compute global basis all a free buying what you'll find that ring him are being able to compute the global basis of this is you what you'll and and then lifting.
the transformation between those who'll the basis that the function if the s. t. which is implemented in the most recent publication of single or e's four one three containing or let the place and it can be found on of this link. a second those serve there are looking to basics which targets fundamental applications of global basis such this new membership problem computing into sections of light years and many feel so and i will begin with the theoretical part we commence with pulling all new ring.
will be introduced p. and. this will normally ring is all the free market generated by a final third acts of generating set off an ideal with two sided ideal in this pretty normal ring scored a global basis if the leading turn off the set g. generate the same. the idea is the leading terms of the original idea itself so in short wilkie is equal to an ally.
let's that remark your that means or four and i knew to be two sided so we take any element s. but the two said i view then we can ride in terms of the generators the eye so we're mother players from the left and the right and as you can see his sins we can have served different pairs of these. other players and p.. we have another some in there which is circling from the next or one to the eye and we have no control is known that unite us what we can do is so we can kind of expect them with the players on the left and right to attend a product and then octane element p.-i which is another man's. not have to put in on your ring but of the enveloping of iran which is the tenth of product of the pudding only ring with its opposite for the indigenous.
now all we define what a global basis and that's an example so example is the eight year generated by to pulling all news or other two terms six x. y. and four was he. and since these only terms. you immediately see their surgery already little basis but now you might know from a feels that when you have a global basis and you pick any other career element of year deal than there is always an element of looking at bases who's leading term divides the leading term off the other early on. a few we have a counter-example so as the two x. y. or z. and term this clearly contradicts this property and this is not because of than one all meals but because of the cohesion to which is neither the visible a six nowhere for now only if you look away the says. this property a few all the fuels so this the visibility of eating terms not leading long as the leading terms then we call a generating search a strong basis. why not to get the missing elements will get two elements are to pulling all years and g. and years now to loosed leading the norm use such there we get one come all knew which all that's both of them in minimal sense.
and then we use the euclidean property of the ring of collisions rather you just need property principal ring to get relation between the least come in multiple greatest on the rise and the leading positions and then you can define as usual in a sling. romeo that is the final such that the leading terms cancel each other held but we've seen previously in the example that this is not enough so you also need a deepening on you who's leading cause of fishing will be the greatest company buys. note however that in the definition of as and usually no news from this mom you key is not unique to find that you might are infinitely many as a dozen people your muse soul only a theory where earthquakes yuri and those you that erm you have a strong global basis.
this different only if oil is and you pull in a museum uses you and this gives you procedure on how to compute such a strong global basis. but let's first look at an example and do this construction of you putting on years soul no previous example we had six x. y. and four ways he now we can compute the for example g.w. one by multiplying get from the right with the end you from.
the left with x. so we get the same an all new explain the end we obtain this going on you that was missing before so i'm certain that both the problem in the previous example. but actually you can also go on you can for example compute you fully to the ignoring this overlap between the leading no neutral at indy and obtain putting normally have two x. y. wives the and this is certainly not be visible by a four g. or any or by the previous the computer deeply know. you are usually one so he also knew that nuchal the bases and you can all go on and on like this with thirty forty three four five and so on and the pump up the split between the leading norm years with any other normal year old other you that you like and you might get more not produce a book with no means. that you need to get to the local basis.
so to to handle this problem. are you have to be the divide as a new building on years into categories so the first one takes into account any overlap in reveal whether of the nieman or years the corresponding to his relation that you see right you and then you can define. leslie normally us survive the first kind corresponding to the moon or muti and also you can you find you balloon you putting on your first and also corresponding to the moon on your tea and in a way sample this was the pulley one. now we continued with ignoring this all that relation and this is what we do you in case to seoul we pump up the split between the leading norm youths with with an arbitrary more muted other you and then you get invested in all new of second kind corresponding to lose w. and also if you. bring on your soul or could fusions remain the same but this one all knew she changes.
so if we sum up the situation you are in theory you can characterize strong global basis with her stronger of maryland nations for example having unique remind us that reduction and this is very similar to the commuter the case. and also we've seen that all rings and access to the norm years you need to those you putting no news to obtain missing when or years in the global basis. and one to compute compute all those in theory you get a procedure that gives you a strong global basis but since we're in finally many of those as and you're pulling on use of second kind was procedure does not turn it in general. i'm so you are only able to compute up to fix third length of norm years and i'm only in certain situations you can do to mind if a global basis of a strong local basis for input you will be fine. i. years so we also all the in order to reduce the costs of the computation want to be able to produce your and actions of us and you putting on or before we actually computed and reuse and so are we that is to say we need great year for. critical care center a critical chaos we mean a pill which we computer in s.g. putting on so let's look at first some quite sure you're there so we generalist from the commuter the case or that require a completely new adjustments so this first on my first one right years server other easy.
to a very high and it tells you that you putting on new reduced to zero live or one needing good vision to devise the other so this could union the proof is only on the level of course visions and there is nothing to considerable the norm years so this is just a generalization of the commuter the caves or the into just now you. i wonder if there's a similar but your e m four is fully no news and the from the few cases from the commute case you might know the productivity room with those you that nestle norm you have a critic of their uses the euro if a leading the norm year cycle plan in the sense that thursday ethanol.
i'm they only have truly are or what relations. and turn on the other hand in the ring case if we assume as well that the middle of the call prime and additionally the leading cause of physicians all of the into your source of corporate and then we can follow the idea of the proof and to write on yesterday normal at and t. like this and simplify but because search. you we are. if we know we'll second kind. and because of this another you with which are comes up the space between the lehman only as we are not as in the future as well as in the commute of caves able to say that is going on you always will reduce the euro so this let you in can it not be or journalist. another used to put you on your category of criteria is so put your arm so let's take three pulling only as you which are satisfy several that relations the last year and then you can say that you've the leading.
inquisition and offered the middle one is to devise the least come in multiple of the leading cause of visions of the other two and if we additionally assume that the us during meals of f one if to end of two or three were used to zero then this will also old for the us for the norm you have one and three. and it was the for the building only of where that now is the leading go fishing office to you by it's the greatest can devise a of the other two leading good visions then because funding you putting all your f one and three we also reduced to zero. turkey in mind however that this is not a general statement as it might seem you because oil is and you pulling all meals are. can your first the second kind but more importantly they are related to a fixed menorah saw this only costs very certain case and this are wrapped up the theoretical part and all my colleagues will continue with practice. for example.
for every year. i agree.
now a fine of course there are already controls. we're burgers and explains read. pushovers from her degree radical recording an extra. the great grandchildren. using the same. with the length of sand and to chronicle generations. i. compare now i'm just going to be. here and i want to computer club members of society. but first time in a friendly option recipe for abuse cases just years singular true always root for reduce global basis trials of hair down. weekend there were a couple of days of i can say with me and as the beautiful places in this case. care now or imagined is we have had performed this competition over the feud with the are interesting to think in this case the those two generators is generator and those with generators they would have managed so in this case the by amateur am giving birth. they are the next was that much will i move his non-trivial to caution or another worst as the one nine zero two tours from some module. no i want a computer reputation for us to cross from such are the the classical method by a warrant for their sofas erm to compute call an ambulance. so in this case in particular want to computer calls are the laws say it's true. i am frustrated was and and. in the first component. it puts the first generation. and the. the second component. and then known for every generation jamie we create a generator here in and because it generates and was compiled. but. their looks like this and be interrupted every generation and come back for the actual can look like this so here in the first components with the the to the second component with the warm if you only in the first component. we have to generate as from injuries. market. the us look as good of his work has earned and saved. cultural grassland. it is more reliable source. today. every incredible paid. one by one. p.. first are ok now bigger or the very first was the real component in the first component. and this was the global basis of some idea air. and if you look closely you already see that the if we are computers to coal and oil. and of a it's true again in the idea want change so in this case there's already a already the saturation idea of i.a.s. to and therefore is also a presentation of the two questions of marginal we were looking for. moreover. have i pods to ellis a subset of high.