Tropical Jacobian Conjecture

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Tropical Jacobian Conjecture
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On a Tropical Version of the Jacobian Conjecture
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We prove that, for a tropical rational map if for any point the convex hull of Jacobian matrices at smooth points in a neighborhood of the point does not contain singular matrices then the map is an isomorphism. We also show that a tropical polynomial map on the plane is an isomorphism if all the Jacobians have the same sign (positive or negative). In addition, for a tropical rational map we prove that if the Jacobians have the same sign and if its preimage is a singleton at least at one regular point then the map is an isomorphism.
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[Music] okay so i will talk about what uh what was what happened with the tropical version of the jacobian conjecture so first i will briefly very briefly remind the classical uh jacob and conjecture i would not dwell much on it because there are very good overviews on it with the about including the
very rich and dramatic history uh and uh then i will say a few words about actually give notifications for the tropical settings uh actually repeating some parts of what stefan just did and then i will tell about results uh on the tropical tropical version of jacobian conjecture okay so uh first about let me briefly remind classical so we have a polynomial map uh for the sake characteristic field zero and consider the jacobian of this polynomial memso consisting the matrix consisting of the partial derivatives and this uh the classical jacobian conjecture due to calorie very old source so for 39 that if the jacobian equals to one then f so the polynomial represents [Music] and each inverse is also a polynomial um well there were a lot of results around it i i will just mention a few of them which have a flavor common with what i will talk about uh tropical tropical uh stuff okay so uh for a for an algebraically closed field f uh if f is injective then f is bijective as well so actually the proof due to axe was modern theoretic and it's very easy in few lines and it's a reduction to finite fields using norse telling that and if if you formulate this statement for finite fields it's trivial because in the finite set if there is an injective map it's also a budget split a very easy nice result and one can actually treat a jacob and 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 contra example in fact which shows that the whole issue of jacobin conjecture is very delicate so if you consider the real field as a ground field and uh the contra example shows that assumption that uh just the jacobin is positive is not enough uh so it is the counter example that we have an i not isomorphism with a uh positive with a positive jacobian so the that it's necessary to have the uh deformant equals actually equals the constant so you can consider this constant to be one okay that's all what i wanted to say about the classical
just just to remind the classical uh conjecture and now let me briefly us in introduce annotations about the tropical world okay so the basic object of the tropical algebra is the tropical submarine and it's in endowed with the operations uh which is denoted in this way and say uh the main source of the tropical same rings is the following we take um ordered semi group and then uh it's it's simmering with inherited iterations
uh as the sum for the minimum and the product as the group systemic group operation uh 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 an uh group then we are talking about the tropical semi-skill 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 uh also the division uh in the tropical semi field which is in fact the uh subtraction in the subtraction in the uh group okay the typical examples uh uh say the non-negative non-negative integers uh all the non-negative integers uh with the added infinity uh these are the examples of commutative tropical submarines and the infinity plays the role of the neutral element and then instant zero plays the role of the unity another examples are just uh so they were these were the examples of the semi rings and if we consider not just the non-negative but the integers uh all the integers with the added infinity these are semi fields because they allow allow subtraction so the division division in the tropical tropical division and an example of the non-commutative tropical submarine is the the semi ring of the n by n matrices say over over any of these semi fields say over the with the infinity uh with the usual operation for the matrix multiplication the another important object is the tropical polynomial so to define it with with the common normal which is just the product of a of a variable uh then then a monomial actually it is a uh is in the classical case and the coefficient tropical times the tropical monomials and uh so we can consider this uh its tropical degree is the sum again is the 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 star the monomials and this could be viewed as a classically as the minimum of uh linear functions so a convex conics function convex piecewise linear function and well what is uh so this was uh before was very uh typical and uh similar to uh classical case but what gives us uh the tropical geometry and what is maybe a psychological difficulty usually is understanding is the concept of the tropical zero so we say and uh it has a lot of justifications the definitions that uh we say that is the tropical zero if uh say of the polynomial of the polynomial f uh if uh the minimum uh is attained at for at least two different values of uh of uh j so for a two different for two different uh tropical monomials so that means that uh in other words if we can see the tropical polynomial as a piecewise linear function then the tropical zero is the uh points at which uh the polynomial is is not smooth uh okay so uh 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 uh this minimum is a tropical polynomial and this is a we can treat as the numerator and the minimum of q's is a denominator and their subtraction uh the difference is the tropical uh tropical fraction and this we can view as a tropical algebraic rational function uh 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 uh and dimensional polyhedra on each of which this function is linear um actually conversely conversely any piecewise linear function can be represented in this form so the is the um is the difference uh difference of two tropical tropical polynomials moreover moreover if you can see the any continuous function it is known that it's a difference of two convex functions so this is a particular case of this uh theorem for uh uh piecewise linear functions uh actually more generally uh and for our considerations would be also valid uh one can assume that the coefficients are real not not just rational uh or integers or more coefficients as it is in the tropical world but we can consider the coefficients to be real so just consider any uh piecewise piecewise linear function and the results would be true but the main question um arises uh how how to replace the jacobian for non-smooth tropical algebraics rational maps so we have now this would be our main object at tropical algebraic rational map so each coordinate is a algebra is a tropical algebraic rational function uh and so how to how to replace the jacobian in say in the jacobian conjecture um well actually uh uh unlike unlike the classical uh situation um the our situation is a little bit uh 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 uh 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 week and one strong version so we start uh with the week version uh and first uh we need the following definition um okay i can see them at a tropical map and in the point and the all the n-dimensional polyhedra containing p on which if it is linear so we know that we it is a tropical map it is piecewise linear so uh we take this uh 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 a1 ak so in the neighborhood in the so that in the neighborhood of the point p so we assume that p is say is in the boundary of uh ck polyhedra on which the map is linear and we can see this map okay and that we take edge codes so the determinants of these matrices and uh uh we consider the convex how of these matrices and denote them by uh dp of f uh so the first uh proposition which is uh uh which uh what i call the uh a weak uh the weak version of the tropical jacobian conjecture that if uh for each uh for each point p uh dp doesn't contain a singular matrix then f is an isomorphism so this is a this dp uh replaces the role of the jacobian so we assume that uh that it doesn't contain a singular matrix that's the um that's the assumption and then we uh state that it's in isomorphism actually the i can give the proof because it is uh it is in it goes in few lines uh it relies essentially on the uh clark's theory that f actually the clark's theorem holds for any lipschitz map is under this uh under this condition so then each dp doesn't contain a singular matrix then it's a local uh 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 moreover for not only for uh 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 [Music] uh so we know already that it's a local home murphys and uh that then because it's proper then it's a global global home memorism so that's the whole proof um okay unfortunately so we have a sufficient condition for a map to be in isomorphism uh but unfortunately it is not necessary and uh i will give a counter example for that which is also quite instructive um 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 triangular and upper triangular isomorphism okay so this is the lower triangle and this is the triangular and if we consider their composition then it is a well it's a it's a linear um in piecewise linear in four pieces and uh on so on four sectors uh 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 uh four uh following uh jacobian matrices well just for from the from the formula for the lower lower triangle and upper triangular isomorphisms and this so if we take the sum of the second and the third matrices uh 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 uh when so when the product when the product is equals to four for example one can take alpha and a equals to zero and beat a then beta equals to two and so this is a counter example which shows that this sufficient condition is is not necessary and it 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 um and the degree on the degree of the map equals equals one uh and the question arises when this condition is sufficient so when the condition of the constancy of the science of of the jacobians is sufficient to to to to be an isomorphism and this is true on the plane when we uh concede the tropical polynomial map so that both f1 and f12 are tropical polynomials that means so they're convex and then indeed that if all the jacobians say are positive then f is an isomorphism but uh beyond beyond these conditions on the plane and for the tropical polynomials this can sufficient sufficiency condition is uh is not necessary condition is not sufficient unfortunately and uh the following example
shows that uh okay i can see the uh the following now we can see the knotted uh polynomial map tropical polynomial but rather a tropical ration map so we write here module function and clearly module function is a is a tropical uh tropical rational function you can easily write it with the with the using the subtraction and we consider the following tropical rational map um it has one can easily verify that it has positive jacobians in all its linear pieces in all pieces where this for the function the map is linear but because this function is central symmetric uh it is not an isomorphism one can modify uh slightly modify this example uh to get now a tropical polynomial map rather than rational but in the three-dimensional space with all positive with all positive jacobians and so being again being not an isomorphism so we see that uh 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 um okay but one can uh formulate a one can simulate a necessary fission 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 uh verify whether a tropical map is an isomorphism so uh i remind that um actually this is uh this definition is uh holds for any for any uh for any map so we say that the point is regular if um for any uh so it's the point in the in the uh target um if for any pre image from from for from this point its jacobian is not zero okay so this then this point is regular and then the by far set of regular points uh the same sorry verbalius is dense okay and then we can simulate now at uh simultaneously and necessarily in 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 theory meant 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 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 uh okay then uh if we take uh any point uh in the target which is out of uh 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 uh 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 the isomorphism and all all this can be performed have an algorithm uh which tests uh whether a tropical map is in fact a in isomorphism okay and uh another issue which is uh related to the jacobin conjecture is the tameness uh of the of the automorphisms uh and it is a dick smear the classical dick smear problem and what do we have in the tropical world in the tropical world um similar again to the classical but uh setting but actually we cannot can we cannot use the statement of the classical uh [Music] in the stating but we can we can prove this independently uh that on the plane uh indeed indeed any automorphism stay what does it mean so we define two classes uh two classes of uh automorphism the first class is triangle actually we had already an example uh before so triangle triangle that means with each we change this triangle we change one only one coordinate at that time and also we can see them an analog of linear tropical rational automorphisms uh so these are automatism all with uh linear and with the determinant uh with the determinant equals plus or minus one and um okay so the proposition uh states that the group uh the group of tropical rational uh homogeneous automorphisms so on the on the plane uh is generated generated by triangle and linear linear automorphism so it is stain in classic in the classical world we know that in the free three-dimensional case the group of automorphism is not tamed so it is not generated by triangle and linear automorphisms here this is an here this is an open question and uh my conjecture is that this is indeed the case the group is is also not same okay thank you thank you very much for for attention thank you lima so are there a question
for lima i i have a question with respect to us proposition so is it related that if we have a convex function then we can uh approximate it by this uh by the sum of uh this triangular functions 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 uh triangle and linear linear automorphisms yeah but but in some sense the same ingredients you have triangular and linea and one you approximate as a sum but here you have exact and and then in multivated case you have to replace triangular by yes yes yes well you can approximate but here it's it's just it is it is an exit it is an ex it is an exact equality of automorphisms it's so therefore for piecewise linear it would be exact i think so if you have uh so if you have just convex function yeah uh then for convex function you have approximation but if you have a linear then it will be exact approximation by the sum of linear and angles yeah again but so great people with such a funny uh level and so here they [Music] you mean the sum but the sum well look um there's some no i don't i i'm not sure well i i 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 i think that i think it is just the opposite this statement is uh i mean it is not tame okay but it's a good point i'll think about the connection yeah because i just hear the same ingredients so therefore yeah yeah and gradients are the same but it's a different statement yeah yeah yeah possible to do with simplexes and then it's possible to estimate but but i think there is a problem so no not only such a triangle you have maybe uh allow some rotation so with exact triangles it's not true in the for for general and then you have i forgot there are no okay so we stopped here we have a prank until 11 50. okay thank you very much thank you glad for the attraction also [Music]