 Okay, so I will talk about what happened with the tropical version of the Jacobian conjecture. So first I will very briefly remind the classical Jacobian conjecture. I would not dwell much on it because there are very good overviews on it including the very rich and dramatic history. And then I will say a few words about actually give notations for the tropical settings, actually repeating some parts of what Stefan just did. And then I will tell about results on the tropical tropical version of Jacobian conjecture. Okay, so first let me briefly remind classical Jacobian conjecture. So we have a polynomial map for the characteristic field zero and consider the Jacobian of this polynomial map. So the matrix consisting of the partial derivatives and this classical Jacobian conjecture due to Keller, very old source of 39, that if the Jacobian equals to one then f, so the polynomial map is nine, isomorphism and its inverse is also polynomial map. Well, there were a lot of results around it. I will just mention a few of them which have a flavor common with what I will talk about tropical stuff. Okay, so for an algebraically closed field f, if f is injective then f is bijective as well. So actually the proof due to ax was model theoretic and it's very easy in few lines and it's a reduction to finite fields using Noosh telling that. And if you formulate this statement for finite fields it's trivial because in a finite set if there is an injective map it's also a bijective. So it's a very easy nice result and one can actually beat a Jacobian conjecture is a local isomorphism. So due to the implicit function theorem applies the global isomorphism and another maybe the second and the last result in this area which I will mention is a counter example in fact which shows that the whole issue of Jacobian conjecture is very delicate. So if you can see that the real field is a ground field and the counter example shows that assumption that just the Jacobian is positive is not enough. So it is the counter example that we have not isomorphism with the positive Jacobian. So that it's necessary to have the job and equals actually equals to a constant. So you can consider this constant to be one. Okay that's all what I wanted to say about the classical just to remind the classical conjecture and now let me briefly introduce the notations about the tropical world. Okay so the basic object of the tropical algebra is a tropical semirin and it's end out with the operations which is denoted in this way and say the main source of the tropical semirings is the following. We take an ordered semi-group and then it's a semiring with inhibited operations as the sum as a minimum and the product as the group semi-group operation. If say you have not just a semi-group but rather a group it could be a billion so always it is ordered then so if it is a group then we are talking about that tropical semi-sq field and if it is a billion so it's a group operation is commutative then we are talking about tropical semi-field and then we can introduce also the division in the tropical semi-field which is in fact the subtraction in the group. Okay the typical examples are say the non-negative integers all the non-negative integers with the added infinity these are the examples of commutative tropical semirings and infinity plays a role of the neutral element and in its turn the zero plays the role of the unity. Another examples are just so they were these were the examples of the semi-rings and if we can see them not just the non-negative but the integers or the integers with the added infinity these are semi-fields because they allow subtraction so the division in the tropical division and an example of the non-commutative tropical semi-ring is the semi-ring of the n by n matrices say over any of the semi-fields say over z with infinity with the usual operation for the matrix multiplication. Another important object is the tropical polynomial so to define it we start with the monomial which is just the product of a variable then a monomial actually it is a it is in the classical case in the coefficient tropical times the tropical monomials and so we can consider its its tropical degree is the sum again is a classical situation the sum of the degree is the sum of the degrees the sum of the powers and classically we can look at this as a linear just as a linear function and well this was a monomial and again similar to that classical case we define tropical polynomial as the tropical sub the monomials and this could be viewed as a classically as the minimum of linear functions so a convex function convex piecewise linear function and well what is so this was before was very typical and similar to classical case but what differs the tropical geometry and what is maybe psychological difficulty usually is understanding is the concept of the tropical zero so we say and it has a lot of justifications the definitions that we say that x is a tropical zero if say of the polynomial of the polynomial f if the minimum is attained at for at least two different values of of j so for a two different for two different tropical monomials so that means that in other words if we consider tropical polynomial as a piecewise linear function then the tropical zero is the points at which the polynomial is is not smooth okay so well we can continue these definitions and extend them to tropical algebraic rational functions so now the tropical fractions and we have seen that the fraction is a subtraction in the classical sense so this minimum is a tropical polynomial and this is a we can treat as a numerator and the minimum of queues as a denominator and their subtraction the difference is the tropical tropical fraction and this we can view as a tropical algebraic rational function and where the p's and q's are linear functions with rational coefficients geometrically it is a piecewise linear function so that means that one can partition the whole space into a finite number of n-dimensional polyhedra on each of which this function is linear actually conversely conversely any piecewise linear function can be represented in this form so as the as the difference difference of two tropical two tropical polynomials moreover moreover if you consider any continuous function it is known that it's a difference of two convex functions so this is a particular case of this theorem for piecewise linear functions actually more generally and for our considerations would be also valid one can assume that the coefficients are real not not just rational or integers so more coefficients as it is in the tropical world but we can consider the coefficients to be real so just consider any piecewise piecewise linear function and the results would be true but the main question arises how to replace the Jacobian for non-smooth tropical algebraic say rational maps so we have now this would be our main object at tropical algebraic rational map so each coordinate is a tropical is a tropical algebraic rational function and so how to how to replace the Jacobian in say in the Jacobian conjecture well actually unlike unlike the classical situation the our situation is a little bit well not a little bit essentially easier because we need to prove only that the map the map is invertible so the inverse does exist because if it does exist then it is also a tropical map so it is a piecewise piecewise linear function and so it can be represented in the same way so we need to test so actually one can view the Jacobian conjecture as the a criterion for a map to be in isomorphism okay what we'll do first I would be no unique version of the tropical Jacobian conjecture but there would be a one weak and one strong version so we start with the weak version and first we need the following definition okay consider a tropical map and the point and all the n-dimensional polyhedral containing p on which if it is linear so we know that we it is a tropical map it is piecewise linear so we take these the pieces on which f is f is linear and for each for each of these for each of these linear maps with which are which are now Jacobian matrices simultaneously we denote them a1ak so in the neighborhood in the so that in the neighborhood of the point p so we assume that p is in the boundary of say k polyhedral on which the map is linear and we consider this map as k and then we take the Jacobians so the determinants of these matrices and we consider the convex hull of these matrices and denote them by dp of f so the first proposition which is which what I call them a weak the weak version of the tropical Jacobian conjecture that if for each point p dp doesn't contain a singular matrix then f is an isomorphism so this is a this dp replaces the role of the Jacobian so we assume that that it doesn't contain a singular matrix that's the that's the assumption and then we state that it's an isomorphism actually the I can give the proof because it is it is in it goes in a few lines it relies essentially on the Clark's theorem that f actually Clark's theorem holds for any Lipschitz map is under this under this condition so then each dp doesn't contain a singular matrix then it's a local homeomorphism so it's a local theorem better to say that if dp for a given p doesn't contain a singular matrix then it's a local homeomorphism and this is true more over for not only for piecewise linear but also for Lipschitz maps then we use the easy observation that tropical map is a proper map so the pre-image of every compact is again compact and this implies that so we know already that it's a local homeomorphism and that then because it's proper then it's a global homeomorphism so that's the whole proof okay unfortunately so we have a sufficient condition for a map to be in isomorphism but unfortunately it is not necessary and I will give a contra example for that which is also quite instructive okay so we can see the tropical map on the plane it would be it would be isomorphism and it's a composition of a lower triangle and upper triangle isomorphism okay so this is the lower triangle and this is the upper triangle and if we consider their composition then it is a well it's a it's a linear in piecewise linear in four pieces and so on four sectors on four sectors of the origin like on the picture and if we consider the d at the origin then it's a convex hull of the four following Jacobian matrices well just for from the formula for the lower triangle and upper triangle isomorphisms and this so if we take the sum of the second and the third matrices we see that it is it is singular when so when the following condition holds so when one of the either beta equals to alpha or b equals to a okay oh sorry when so when the product when the product is equals to four for example one can take alpha and a equals to zero and beta the beta equals to two and so this is a counter example which shows that this sufficient condition is is not necessary and okay still it would be nice to have a necessary sufficient condition uh and so we start with the following is the mark that if a tropical map is an isomorphism then all the Jacobians have the same sign so either all of them are positive or negative this is due to due to orientation and the and the degree on the degree of the map equals equals one and the question arises when this condition is sufficient so when the condition of the constancy of the signs of of the Jacobians is sufficient to to to to be an isomorphism and this is true on the plane when we consider the tropical polynomial map so that both f1 and f2 are tropical polynomials that means so they are convex and then indeed that if all the Jacobians say are positive then f is an isomorphism but beyond beyond these conditions on the plane and for the tropical polynomials this can sufficient sufficiency condition is is not necessary condition is not sufficient unfortunately and the the following example shows that okay consider the following now we consider not a polynomial map tropical polynomial but rather a tropical rational map so we write here module function and clearly module function is a is a tropical tropical rational function you can easily write it with the with the using the subtraction and we consider the following tropical rational map it has one can easily verify that it has positive Jacobians in all its linear pieces in all the pieces where it's the function the map is linear but because this function is central symmetric it is not an isomorphism one can modify slightly modify this example to to get a now a tropical polynomial map rather than a rational map but in the three-dimensional space with all positive with all positive Jacobians and so being again being not a isomorphism so we see that we have a now necessary condition of positivity say of all the Jacobians but it's not it's not sorry necessary condition but it's not sufficient okay but one can formulate one can formulate a necessary and sufficient condition but it's now it looks not not so natural but it's on the other hand it's good to have an algorithm to verify whether a tropical map is an isomorphism so I remind that actually this is this definition is holds for any for any for any map so we say that the point is regular if for any so it's the point in the in the target if for any pre-image from from from this point each Jacobian is not zero okay so this then this point is regular and then by the set of regular points the same sorry values is dense okay and then we can formulate now at simultaneously a necessary and sufficient condition for a tropical map to be an isomorphism namely namely that means that the condition from the previous slide that all the Jacobians have the same sign and now we require that at least for one regular value the pre-image is unique okay well the necessity is trivial but the statement of the theorem and this this condition is sufficient also so if such a point does exist at least one point then the whole tropical map is an isomorphism and relying on this theorem we can now design an algorithm to verify whether a tropical map is an isomorphism namely an algorithm yields a partition into polyhedra such that f is linear on each pi this can be done this can be done by means of linear programming okay then if we take any point in the target which is out of the union of the boundaries boundaries of this polyhedra so we take the boundaries of this polyhedra which are polyhedra of one less dimension and take the images and subtract them then any point out of this out of this union is regular and we can apply the criterion from the previous theorem test test that the pre-image is unique of this point and if it is the case then we know from the previous theorem then we are dealing with isomorphism and all this can be performed have an algorithm which tests whether a tropical map is in fact an isomorphism okay and another issue which is related to the Jacobian conjecture is the timeness of the of the automorphisms and it is a classical Dixmere problem and what do we have in the tropical world in the tropical world a similar again to the classical but setting but actually we cannot we cannot use the statement of the classical in the setting but we can we can prove this independently that on the plane indeed indeed any automorphism is staying what does it mean so we define two classes two classes of automorphism the first class is triangle actually we had already an example before so triangle triangle that means with the we change this triangle we change one only one coordinate at a time and also we consider an analog of linear tropical rational automorphisms so these are automorphism all with linear and with a determinant with a determinant equals plus or minus one and okay so the proposition states that the group the group of tropical rational homogeneous automorphisms so on the on the plane is generated generated by triangle and linear automorphism so it is tame in classic in the classical world we know that in the three-dimensional case the group of automorphism is not tame so it is not generated by triangle and linear automorphisms here this is an here this is an open question and my conjecture is that this is indeed the case the group is also not tame okay thank you thank you very much for for attention thank you so I have a question with respect to last proposition so is it related that if we have convex function then we can approximate it by the by the sum of this triangular function oh no I don't think so because this is not approximation this is an exact statement you you have a automorphism and you need to represent it as a product of triangle and linear linear automorphisms yeah but but in some sense of the same ingredients you have triangular and linear yeah in one you approximate as a sum but here you have exact and and then in multivariate case you have to replace a triangular by simplixes yes yes yes well you can approximate but here it's it is it is an exact it is an it is an exact equality of of automorphisms it's because you have a function at piecewise linear yeah so therefore for piecewise linear it would be exact I think so if you have so if you have just convex function yeah then for convex function you have approximation but if you have linear then it would be exact approximation by the sum of linear and at angles yeah again but many people have this lemma but it's called hardy little wood and someone so it is funny funny so great people with such a funny lemma and so here they said if you have piecewise linear then it would be exact you can just obtain this piecewise linear function as a sum I don't know I'm well you mean the sum but the sum well look the sum no I don't I I'm not sure well I agree that it's related but I'm not sure that it will give the exact result well it's a good point I will think but I don't think that I think that is I think it is just the opposite the statement is I mean it is not tame okay but it's a good point I'll think about the connection yeah because I just see the same ingredients so therefore yeah yeah ingredients are the same but it's a different statement yeah yeah and in general it's possible to do with seplexes and then it was supposed to estimate but but I think there is a problem so no not only such a triangle you have maybe allow some some rotation so with exact triangles it's not true in the for for for general and then you have I got exactly but but I can't find the difference oh yeah okay let's send me then the link yeah that would be interesting thank you for the nice talk okay oh there are no patients so we stop here we have a practice until 1150 okay thank you very much thank you black for the interaction also