Real tropical hyperfaces by patchworking in polymake
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Real tropical hyperfaces by patchworking in polymake

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CC Attribution 3.0 Germany:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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2020

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English

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2020

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Berlin

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Abstract 
Hilbert's 16th problem asks to classify the isotopy types of real algebraic hypersurfaces in projective space. In the 1980s Viro developed patchworking as a method to construct real algebraic hypersurfaces with unsually large mod 2 Betti numbers. Interpreted within the larger framework of tropical geometry, patchworking leads to a combinatorial approach to real tropical hypersurfaces. We report on a recent implementation of this method in polymake, we compare with a previous implementation of de Wolff et al. in Sage, and we report on experiments with surfaces of degrees up to 6.

Keywords  real algebraic curves and surfaces patchworking tropical geometry mathematical software Hilbert's 16th Problem 
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00:00
but i and my name is michelle u.s. fish and today i will speak about real tropical hyper surfaces by page working in italy make.
00:12
and this is short walk with paul fatah from the i liked.
00:20
a the setup is as follows in tropical geometry we start the tropical pulling on gozo tropical hyper surfaces them by out policy will geometry methods so what's the idea well there's an area's ways to describe this and i will take one example so here i have a political meal which depends which has. three variables x. y. and z. and it is a homogeneous pulling on your off degree three with ten terms altogether. and so in the quote physicians there's a permit a t. which is a positive real number but it's supposed to be small and i have some various powers of tea. equipped with science as cliff issues so i may read as a real pulling on now one way to get at a tropical geometry is via the procedure of mass lofty condensation and this means taking such expressions taking the longer of them with respect to the two are the basis of tea. and then drive the permit a t two zero. in this way we would arrive at the in this particular case at the tropical pollen on your which is a minimum of ten terms one for each of the terms of the original pollen normal and for instance the this x. three hour by hour longer them speak comes three times x. or hear this the t. minus. the to the seven why is he square becomes seven plus why the two times the please note that the quick efficient as the play a role only as far as the the exponent of the lowest or the the matters and in particular the science do not play. the role of or right. so i'm such a tropical pollen normalise no to vanish if that minimum is attained at least twice and then there was general results the fundamental of theorem of tropical john a tree relating the the other by hyper surfers defined by the classical pollen on your with a very strong local. those which is then defined as the tropical hyper surfaces surface of this tropical putting on our right and so the background you could explain it in terms of valuation theory if you like. the benefit is that now because everything becomes a piece by senior police room geometry method supply and in particular you cut out of the most a comic side computations and similar the which you can now use in order to us to study them outright geometry. the was patch working is an idea which the pre the dates tropical geometry goes back to the nineteen eighty's. and this is about an hour to doing so in this new tropical john to language i could say this is the same procedure as before but now we also want to refinance information by taking the science of this real pollen on your into account in order to describe the real costs. so what i'm reporting on here there's a new implementation of this the comet tory a patch working in the public. all right so let's look at ing the example so this is actually the tropical have a surface year to the left him the which comes about from the from the previous pulling on earth so it had ten terms and each one of these terms corresponds to one of these regions so for instance a this region he. here in the middle. response to them the term with him ok let's let's look it up the so it is this time or here and the so here we have three plus exploits why plus z. and so everybody in the.
04:19
this hexagon is a point where. the minimum is attained at that particular term.
04:27
the likewise for all the other guys so them we have these these signs so here in this or original pollin on your own the this has a negative sign so this is this the minus over here.
04:42
and the negative sides are indicated with a minus here and likewise the positive sciences year old we have ten terms each have one of them with a sign him the black lives started on not our form the tropical happen surface and this but case of plain tropical core of them which.
05:02
the is a tropical car philip the curve of degree three and so what patch working tells you the the real tropical curve is something which comes about from the. is a part of the surface which sits in between regions of opposite sides of this indicated handled but there's a little bit more which is needs to be taking into account and this is visible from becomes visible by looking at the at the do a picture so everything can be described in terms offered you will stop divisions. of the support and so these ten times correspond to let us points in the plane and saw the corporations can be seen as height functions looking at law context was defines a. the regular subdivision in this case it's the tri nations so whenever the conditions are generic enough then it's the triangulation like in this case this is due will to this hyper suffer so it contains exactly the same information and now these a this really low costs also shows up in this setting off. over here are so this means that the in these regions these things that we see here correspond to that part over here and then there's a rule and this is what the patch working the actually describes.
06:33
i'm which takes a this one triangulation takes two to the and copies which are reflected along the court and high plains and the room then you need to take some parity a information into account in order to reflect the science properly and then. what you get in this space which is actually a real project of pain. then this blecker of that you get is a piece was leading a curfew which is the same close enough to describe the. topology the correct isotope he tied off the original cause of that we started out with the real curve that we started out with his home or two p. isotopic to this black piece was in your car that we said he and that's to say a version off the patch working here and you to feel now. so what's the purpose of this particular example so this is a hard core of the because i'm hanukkah in at the end of the nineteenth century prove the theorem which says that the number of connected components of a real plane algebra occur of have given degree is at most a certain. question which is quite dramatic in the inn in the degree and if you and i'll do the math for de equals three then that formula equals evaluates to two. so this means a. for off degree three project of real or buy coverage degree three the can have at most to connect with components and he will see an example here this as one of the components of this continues like here and here so this is one component and the other component us. there's all this piece was leaning over over here so this is a core of which was which maximises this balance due to heart this particular the result from the contributions to him. the the hill but the asking for the following task so which could now be summarised as classify the isotope the types of three other bike so the curtain surfaces or of course of the four hundred brian varieties hyper surfaces of higher. stuff i had imagined but that is already a concours and services already complicated enough so what you see here is an original is a snippet off the original text of his about from nineteen hundred and here you see that he explicitly mentions hierarchy. so this is actually the reference all right so what happened to since nineteen hundreds quite a lot but the problem as it stands is not completely solved yet a lot is known and i am totally i'm just by emitting many many great results but let me just highlight if you. so be all completed or classification oft of eyes of two p. types of curves of degree at most seven. the column have been classified surfaces off degree at most a four and saw both of these results build on a lot of previous work by many many many other see other people in particular the researchers from the russian school. another bright geometry. important for our context is the work of fuel which was mentioned several times and later he married and he they develop the method of patch working it's a common a tory other gadget to construct religious right cross with interesting properties that's the goal so there's a famous a. projects are off a former students offer of a hill birds a surge in iraq stereo that was disproved along the way and if you want to read about the more reason to count on the on the history year and about what is known so i recommend with the work of paper by they don't need all in shock. or right so what did we do so i already said so we had this implementation we have this implementation policy making and we now provide for the first time as i as i think the census have quite a substantial number of the off the top a logical information about patchwork so. offices so we investigated decrease three up to six and are so this becomes more complicated with higher degree i will mention something about this a little bit later and therefore we take more examples follower degrees and the we take the the random. regular trying relations which are dual to these the tropico high for services complex hyper services if you wish and the then add science in order to get the information necessary for the real tropical at the services we pick those at random and then i'm we see what the topology us that we get. and the topology is measured in terms of seem odd to betty to us and princes let me explain this so this is the dire i'm on the left here this is a four d. equals three.
12:06
so it can be shown that for such i have a surface the it always has one or two components one component is marked in in blue so this is a more frequent and then the second betty number so this is the number that you see here as as so baby one. it is shown here and so for instance this bar that shows how many of the services that we created has baby want equal to three and betty zero equal to one being connected and so you see you see the distribution notice this is a lock scale here and the source. sometimes if you're lucky. then you get the also disconnected services which have exactly two connected components and them necessarily better one equals one is ok and the situation for the grief for is a little bit more complicated the there are theoretical bonds known for the baby. the numbers of patchwork surfaces for instance to to even back and and other people and to look it up and the paper off we don't watch all for instance. and the so then four degree three year degree for the census that we have here is complete in the sense that all the possible cases occur services. this is the first observation which comes out of this. so let me point out so there's something that i reported on the previous i c m s so we develop software and george work with script jordan and last customer in the top form to enumerate regular trying nations and that we can use now in order to the tube to produce sufficiently many of them patch. work surfaces. in particular for the degree three for the q.b. case we were able to enumerate all regular the of and dense trying relations so using all the letters points these are twenty month million and these served as a basis for the computation of this diagram over here are ok. so now a few words concerning the implementation itself and policymakers not the first implementation of this topic so there is a come out royal patch working to offer you have as our own coffers which works for curse them it is for primarily for visualisation purposes and then there's your sage by team would have all. open courses and this is what we compare with here. so as you can see so there's a this is to experiments first farm to the left is on curves and the rest of the other one is on surfaces and both cases the x. axis is a degree and the y. axis is is the timing again the timings are given the block scale. to see that i'm pulling make is his way faster. so for corus and four surfaces and that's why we can for the first time computer make these computationally in large numbers didn't show you the computations forty he was five and he quit six because it's more complicated to explain what what what you will get but this isn't the paper so let me bring quickly point. guards what's the actual algorithm that is implemented so the importers a tropical pollen on your undersigned back to our very computer regular triangulation of the supporters so that's a context are computation then we directly compete with the chain complex with easy to court officials and finally the betty numbers are determined by our costs elimination of as the tool so this is more or less the same. same as what to view was age does except for so here we take a shortcut to solve your sage contracts are quite explicitly construct a simpler complex which we don't this is one of the reasons why we are fast and let me come to the conclusion from the soul. and now we implement viewers to comment or a patch working in palm eric which is fast enough to compute really many example so in the hundred thousands and millions. so i should also say so what i'm saying is available in the most recent version of pollen make which is version four point one which is a few weeks all by now and actually since we rode up the paper which is in the proceedings we still worked on the thing and the so the current implementation which is already available is. more general then what we rode up in the paper so we also allow for non regular patch working now so this goes beyond tropical geometry and is interesting for common troy reasons but that's another story so our end with this beautiful example over here which is a real tropical cubic surface with two connected cupolas which you can. it recognised by color. and sitting in our p three. with this i would like to end the talk thank you for your attention.