aa r X i v : . [ m a t h . AG ] M a r Decomposing tropical rational functions
Dima GrigorievCNRS, Math´ematique, Universit´e de Lille, Villeneuve d’Ascq, 59655, Francee-mail: [email protected]: http://en.wikipedia.org/wiki/Dima Grigoriev
Abstract
An algorithm is designed which decomposes a tropical univariaterational function into a composition of tropical binomials and trino-mials. When a function is monotone, the composition consists just ofbinomials. Similar algorithms are designed for decomposing tropical al-gebraic rational functions being (in the classical language) piece-wiselinear functions with rational slopes of their linear pieces. In addition,we provide a criterion when the composition of two tropical polynomialscommutes (for classical polynomials a similar question was answered byJ. Ritt).
Introduction
We study decomposing tropical univariate rational functions (compositions oftropical rational functions find applications in deep learning of neural networks,see e. g. [7]). A tropical rational function is the tropical quotient (which cor-responds to the subtraction in the classical sense) of two tropical polynomials.Thus, a tropical rational function is (classically) a piece-wise linear functionwith integer slopes of its linear pieces. A tropical root of a tropical rationalfunction is defined as a point at which the function is not differentiable.Relaxing the requirement that the slopes are integers allowing them to berationals, we arrive to the concept of tropical algebraic rational functions ortropical Newton-Puiseux rational functions [3] playing the role of algebraicfunctions in tropical algebra.In classical algebra the problem of decomposing polynomials, rational andalgebraic functions was elaborated in [1], [5], [2]. In tropical algebra the answerto the decomposing problem differs essentially from its classical counterpart.1e show that a tropical rational function is a composition of binomials andtrinomials. The similar holds for tropical algebraic rational functions.In section 1 we introduce tropical monotone rational and algebraic rationalfunctions and bound the number of tropical roots of compositions of tropicalpolynomials, monotone rational functions and rational functions.In section 2 we design an algorithm which decomposes a tropical alge-braic function and also a tropical monotone algebraic rational function into acomposition of tropical binomials. In addition, we design an algorithm whichdecomposes a tropical algebraic rational function into a composition of trop-ical binomials and trinomials. Moreover, we provide a bound on the numberof composants.In section 3 decompositions of tropical rational functions (so, with integerslopes of their linear pieces) are studied. We design an algorithm which de-composes a tropical monotone rational function into a composition of tropicalmonotone binomials and monotone trinomials. Also we design an algorithmwhich decomposes a tropical rational function into a composition of tropicalbinomials and trinomials. In addition, a criterion is provided, when a trop-ical monotone trinomial is decomposable. Finally, bounds on the number ofcomposants are given.In section 4 we prove that the composition of two tropical polynomials f, g without free terms commutes: f ◦ g = g ◦ f iff there is a common fixed point x (perhaps, x = ∞ ) for both f, g , i. e. f ( x ) = g ( x ) = x and there exist atropical increasing algebraic rational function h and integers a, b ≥ , k, m ≥ f = h k , g = h m or f = ax + x (1 − a ) , g = bx + x (1 − b ) onthe interval ( −∞ , x ] (similar conditions hold on the interval [ x , ∞ )), unless f = x + c , g = x + c , x ∈ R for some c , c ∈ R . Also we provide an exampleof a pair of increasing tropical rational functions with commuting compositionwhich do not satisfy the latter conditions. For classical polynomials the answerto commutativity was given in [9], [10] (more recent generalizations and furtherreferences one can find in [8]), in which commuting Chebyshev polynomialsplay a crucial role.In section 5 we introduce tropical polynomial (respectively, Laurent poly-nomial and rational) parametrizations of polygonal lines. We show that anypolygonal line admits a tropical rational parametrization and provide criteriawhen it does admit a tropical polynomial (respectively, Laurent polynomial)parametrization. Recall (see e. g. [6]) that a (univariate) tropical polynomial has a form f =min ≤ i ≤ d { ix + a i } , a i ∈ R ∪ {∞} . Linear functions ix + a i , ≤ i ≤ d are calledtropical monomials. So, the minimum plays the role of the addition in tropical2lgebra, while the addition plays the role of the multiplication. Thus, f is aconvex piece-wise linear function with integer slopes of the edges of its graph(sometimes, slightly abusing the terminology we call them the edges of f ). Weconsider the natural ordering of the edges from the left to the right. A point x ∈ R is a tropical root of f if the minimum in f is attained at least for twolinear functions ix + a i , ≤ i ≤ d . In other words, tropical roots of f are thepoints at which f is not differentiable.A tropical rational function is a difference (which plays the role of the di-vision in tropical algebra) of two tropical polynomials. It is a piece-wise linearfunction. So, its graph consists of several edges. Conversely, any continuouspiece-wise linear function with integer slopes of its linear pieces (edges of itsgraph) is a tropical rational function (cf. [3] where one can find further refer-ences). As the roots of a tropical rational function we again mean the pointsat which the function is not differentiable.If g, h are tropical rational functions with p, q tropical roots, respectively,then (see [4]) the number of the roots of • min { g, h } is at most p + q + 1; • g + h or g − h is at most p + q .In this paper we study compositions g ◦ h (being tropical rational functionsas well). Note that if g, h are tropical polynomials then g ◦ h is also a tropicalpolynomial. If s , . . . , s k are consecutive (integer) slopes of (the linear piecesof) a tropical rational function g then g is a tropical polynomial iff s > · · · >s k ≥ monotone increasing (or decreasing, respectively) rationalfunction g its slopes are positive (respectively, negative). Note that any trop-ical polynomial min ≤ i ≤ d { ix + a i } without free term is monotone increasing.One can directly verify the following proposition. Proposition 1.1 If g, h are tropical monotone rational functions with p, q tropical roots, respectively, then the tropical monotone rational function g ◦ h has at most p + q tropical roots. Moreover, if on an interval [ a, b ] ⊂ R function h is linear with a slope s , and g is linear with a slope l on the interval [ h ( a ) , h ( b )] (respectively, [ h ( b ) , h ( a )] ) when h increases (respectively, decreases) then g ◦ h is linear on the interval [ a, b ] with the slope sl . Remark 1.2
In general, the number of tropical roots of the composition g ◦ h of tropical rational functions does not exceed pq + p + q . Moreover, if s , . . . , s p (respectively, t , . . . , t q ) are the slopes of (the graph of ) g (respectively, h ) listedwith possible repetitions (multiplicities), then the slopes of g ◦ h are among s i t j , ≤ i ≤ p, ≤ j ≤ q .For a tropical rational function g = min { x + 1 , − x + 1 } the number of thetropical roots of k iterations of g k := g ◦ · · · ◦ g is k − [7] (see also [4]). i { b i x + a i } , ≤ b i ∈ Q , we arrive tothe concept of tropical algebraic functions (or tropical Newton-Puiseux polyno-mials ) [3]. Respectively, we consider tropical algebraic rational functions beingdifferences of tropical algebraic functions [3]. Remark 1.3
The above statement in Proposition 1.1 on the slopes of trop-ical rational functions holds for tropical algebraic rational functions as well withthe difference that now we admit rational slopes rather than just integers. Theabove bounds on the number of tropical roots also hold literally for tropicalalgebraic rational functions.
In this section we consider tropical algebraic rational functions. As a tropical algebraic rational binomial we mean a function of the form eithermin { b x + a , b x + a } , = b , b ∈ Q or max { b x + a , b x + a } . In thegeometric language the former function is a convex piece-wise linear functionwith two (unbounded) edges (and we call it a tropical algebraic binomial ), whilethe latter one is concave. If b , b > Proposition 2.1 (i) There is an algorithm which for a tropical algebraic func-tion f with k tropical roots yields a decomposition of f into k tropical algebraicbinomials;(ii) let f be a tropical monotone algebraic rational function with k tropicalroots. Then the algorithm yields a decomposition of f into k tropical monotonealgebraic rational binomials. Remark 2.2
Due to Proposition 1.1 and taking into the account that eachtropical algebraic rational binomial has a single tropical root, we conclude thatin Proposition 2.1 one can’t take less than k composants. Proof . The proofs for both items (i), (ii) proceed similarly. Let f haveconsecutive slopes s , . . . , s k of its linear pieces. Recall that s > s > · · · >s k ≥ s , . . . , s k > x l the l -th tropicalroot of f, ≤ l ≤ k . Take a (piece-wise linear) function h with k slopes s .s , . . . , s l − , s l +1 · s l − /s l , . . . , s k · s l − /s l coinciding with f for x ≤ x l and replacing f by the composition with thelinear function (( s l − /s l ) x + f ( x l )(1 − s l − /s l )) ◦ f for x ≥ x l . Thus, h has the4ropical roots x , . . . , x l − , x l +1 , . . . , x k . The described procedure replacing f by h we call straightening : one tropical root (at x l ) disappears.Take a tropical algebraic rational binomial g coinciding with the identityfunction x → x for x ≤ f ( x l ) and with the linear function ( s l /s l − ) x + f ( x l )(1 − s l /s l − ) for x ≥ f ( x l ). Then f = g ◦ h . Note that in case (i) g is a tropicalalgebraic binomial since s l − /s l < k we complete the proof of the Proposition. ✷ Remark 2.3
Observe that each tropical root of f corresponds to a suit-able composant in a decomposition of f . Thus, by choosing (in the proof ofProposition 2.1 above) the tropical roots in different orders, we obtain k ! ”com-binatorially different types” of decompositions of f . Now let f be a tropical algebraic rational function. Let for definiteness thefirst edge of f with a non-zero slope have a positive slope. Consider tropicalroots x of f such that f has an edge with a negative slope to the right of x . Ifthere does not exist such x then f is (non-strictly) monotone increasing, andwe proceed to study the monotone case later. Among such x pick x (perhaps,if not unique then pick any of them) with the maximal value f ( x ). Take atropical root x > x of f with the minimal value f ( x ). We have f ( x )
0, while the edge to the right from x has a negative slope s < x ). Take as g a tropicalalgebraic rational binomial which coincides with the identity function x → x for x ≤ f ( x ) and with a linear function ( s /s ) x + f ( x )(1 − s /s ) for x ≥ f ( x ).So, g is a tropical non-monotone algebraic rational binomial.As h take a tropical algebraic rational function which coincides with f for x ≤ x and coincides with the composition with the linear function (( s /s ) x + f ( x )(1 − s /s )) ◦ f for x ≥ x . Then f = g ◦ h . By a block of edges of f we mean a sequence of consecutive edges of the equal signs of their slopes(ignoring edges with zero slopes). Observe that h has one less block of edgesthan f does. Thus, by passing from f to h we straighten f at point x .Otherwise, if one of adjacent to x edges has zero slope then as g take atropical binomial coinciding with the identity function x → x for x ≤ f ( x )and with the linear function − x + 2 f ( x ) for x ≥ f ( x ). As h take a tropicalalgebraic rational function which coincidies with f for x ≤ x and with thecomposition with the linear function ( − x + 2 f ( x )) ◦ f for x ≥ x . Then again f = g ◦ h , and h has one less block of edges than f does. On the other hand, h has the same number of tropical roots as f does, so one does not straighten a5iece-wise linear function at a point if one of two adjacent edges to this pointhas zero slope.Now we proceed to the case when f ( x ) takes a value greater than f ( x ) forsome x > x . Then min { f ( x ) : x ≥ x } = f ( x ).A tropical regular algebraic rational trinomial is a piece-wise linear functionwith 3 edges having rational non-zero slopes. If the slopes are decreasing orincreasing positive integers we talk about a tropical trinomial .Construct the following tropical algebraic rational functions h, g . If boththe edge of f with the right (and respectively, the left) end-point ( x , f ( x )) hasa non-zero slope s + (respectively, a non-zero slope s − ) then h on the interval( −∞ , x ] coincides with the composition ( − ( s − /s + ) x + f ( x )(1 + s − /s + )) ◦ f . Note that s + > , s − <
0. As g take a function which on the interval( −∞ , f ( x )] coincides with the linear function − ( s + /s − ) x + f ( x )(1 + s + /s − ).We have max { h ( x ) : x ≤ x } = f ( x ) and g (( −∞ , f ( x )]) = ( −∞ , f ( x )]. Incase if s + · s − = 0 then h on the interval ( −∞ , x ] coincides with f , and g onthe interval ( −∞ , f ( x )] coincides with the identity function x → x .On the interval [ x , x ] the function h in both cases coincides with the com-position ( − x +2 f ( x )) ◦ f , and g on the interval [ f ( x ) , f ( x ) − f ( x )] coincideswith the linear function − x +2 f ( x ). Then h ([ x , x ]) = [ f ( x ) , f ( x ) − f ( x )]and g ([ f ( x , f ( x ) − f ( x )]) = [ f ( x ) , f ( x )].Finally, define h on the interval [ x , ∞ ) and g on the interval [2 f ( x ) − f ( x ) , ∞ ). Similar to the consideration above of the interval ( −∞ , x ] denoteby t − (respectively, t + ) the slope of the edge of f with the right (respectively,the left) end-point ( x , f ( x )). If t − · t + = 0 (in this case t − < , t + > x ) then h on the interval [ x , ∞ ) coincides with the compositionwith the linear function ( − ( t − /t + ) x + 2 f ( x ) + f ( x )( t − /t + − ◦ f . In thiscase g on the interval [2 f ( x ) − f ( x ) , ∞ ) coincides with the linear function − t + /t − x + f ( x ) + t + /t − (2 f ( x ) − f ( x )). Then min { h ( x ) : x ≤ x < ∞} =2 f ( x ) − f ( x ) and g ([2 f ( x ) − f ( x ) , ∞ )) = [ f ( x ) , ∞ ).Otherwise, if t − · t + = 0 then h on the interval [ x , ∞ ) coincides with thecomposition ( x +2 f ( x ) − f ( x )) ◦ f , and g on the interval [2 f ( x ) − f ( x ) , ∞ )coincides with the linear function x − f ( x ) + 2 f ( x ). In this case againmin { h ( x ) : x ≤ x < ∞} = 2 f ( x ) − f ( x ) and g ([2 f ( x ) − f ( x ) , ∞ )) =[ f ( x ) , ∞ ). Thus, f = g ◦ h .Observe that h has two less blocks of edges than f does. Also note that x is a tropical root of h with the adjacent to x edges of h with the equal signsof their slopes iff x is a tropical root of f satisfying the same property (we callsuch x a non-extremal tropical root of h because x is not a local extremal of h ). In addition, the numbers of edges with zero slope are the same for f andfor h .Thus, applying two described decomposition procedures to f and obtaining g to be either a tropical non-monotone algebraic rational binomial or a tropical6egular algebraic rational trinomial, while it is possible, we arrive to a tropicalalgebraic rational function f which is non-decreasing, so the slopes of its edgesare non-negative. Thus, f = g ◦ · · · ◦ g k ◦ f (1)where each of g , . . . , g k is either a tropical non-monotone algebraic rationalbinomial (their number among g , . . . , g k denote by k ) or a tropical regularalgebraic rational trinomial (their number denote by k := k − k ). Therefore, k + 2 k equals the number of blocks of edges of f .Now take a non-extremal tropical root x of f . Let s − > s + >
0) be the slope of the adjacent to x left edge (respectively, right edge) of f . Denote by g (1) a tropical monotone increasing algebraic rational binomialwhich coincides with the identity function on the interval ( −∞ , f ( x )] andwhich coincides with the linear function ( s + /s − ) x + f ( x )(1 − s + /s − ) on theinterval [ f ( x ) , ∞ ). Denote by h a tropical non-decreasing algebraic rationalfunction which coincides with f on the interval ( −∞ , x ] and which coincideswith the composition (( s − /s + ) x + f ( x )(1 − s − /s + )) ◦ f on the interval [ x , ∞ ).Then f = g (1) ◦ h and h has no tropical root at x , while having all othertropical roots of f , so h is a straightening of f . Applying the just describedprocedure to all non-extremal tropical roots of f , we obtain a decomposition f = g (1)1 ◦ · · · ◦ g (1) k ◦ f (1) (2)where each of g (1)1 , . . . , g (1) k is a tropical increasing algebraic rational binomial.Every second edge of f (1) has zero slope, and the number k of edges withzero slope of f (1) equals the same number of f . Observe that k equals thenumber of non-extremal tropical roots of f (and also equals the number ofnon-extremal tropical roots of f ). Hence k + k + 2 k does not exceed thenumber of edges with non-zero slopes of f .As a tropical singular algebraic rational trinomial we mean a trinomialwhose middle edge has zero slope. Slightly abusing the terminology, we admitsingular trinomials without one or two edges with non-zero slopes.We are looking for a decomposition f (1) = g (0) k ◦ · · · ◦ g (0)1 (3)where g (0) i , ≤ i ≤ k is a tropical singular algebraic trinomial. To decomposetake the left-most interval [ x , x ] on which f (1) is constant, in other words,the edge of f (1) on [ x , x ] has zero slope. It can happen that x = −∞ , inthis case some of the following considerations become void. Define g (0)1 on theinterval ( −∞ , x ] as the identity function and on the interval [ x , x ] as theconstant function with the value x . 7et f (1) on the interval ( −∞ , x ] equal a linear function sx + r (so, s is the slope of the left-most edge of f (1) ), in particular sx + r = f (1) ( x ).Define g (0) (later we’ll get that g (0) = g (0) k ◦ · · · ◦ g (0)2 ) on the interval ( −∞ , x ]as the linear function sx + r . Therefore, g (0) ◦ g (0)1 on the interval ( −∞ , x ]coincides with f (1) . The same coincidence holds on the interval [ x , x ] aswell. Let the edge of f (1) with the left end-point ( x , f (1) ( x ) = sx + r )have a slope p . Then define g (0)1 on the interval [ x , ∞ ) as the linear function( p/s ) x + x − ( p/s ) x . Also define g (0) on the interval [ x , ∞ ) as the composition f (1) ◦ (( s/p ) x − ( s/p ) x + x ). Then f (1) = g (0) ◦ g (0)1 .Now we observe that g (0) is a continuous non-decreasing piece-wise linearfunction: we have constructed it by gluing at x two non-decreasing piece-wiselinear functions both having the value sx + r = f (1) ( x ) = f (1) ( x ) at x .Moreover, the slope of the edge of g (0) with the right end-point ( x , f (1) ( x ))equals s which coincides with the slope of the edge of g (0) with the left end-point ( x , f (1) ( x )). Therefore, g (0) has no tropical root at x (so, g (0) is astraightening of f (1) ), and g (0) is of a similar shape as f (1) , i. e. g (0) is anon-decreasing piece-wise linear function whose every second edge has zeroslope. On the other hand, g (0) has one less edge with zero slope than f (1) does.Continuing in this way, we construct a required decomposition (3).Combining (1), (2) and (3) we complete the proof of the following theorem. Theorem 2.4
There is an algorithm which decomposes a tropical algebraicrational function f = g ◦ · · · ◦ g k ◦ g (1)1 ◦ · · · ◦ g (1) k ◦ g (0) k ◦ · · · ◦ g (0)1 (4) where each g i , ≤ i ≤ k is either a tropical regular algebraic rational trinomialor a tropical non-monotone algebraic rational binomial (cf. (1)), each g (1) j , ≤ j ≤ k is a tropical monotone algebraic rational binomial (cf. (2)), and each g (0) l , ≤ l ≤ k is a tropical singular algebraic trinomial (cf. (3)).Moreover, if k is the number of tropical regular algebraic rational trinomi-als, and k is the number of tropical non-monotone algebraic rational binomialsin (4), so k + k = k then k + k is the number of blocks of edges of f of theequal (non-zero) signs of their slopes. The number k + k + k does not exceedthe number of edges of f with non-zero slopes, finally k equals the number ofedges with zero slopes. Remark 2.5
The number of tropical roots of f is greater or equal to k +2 k , and on the other hand, is less or equal to k + k + k + k , the latternumber also equals the total number of tropical roots in the composants of f from (4) (cf. Remark 1.2). Tropical rational functions
In this section we study decompositions of tropical rational functions, we recallthat the slopes of edges of a piece-wise rational function f are integers (unlikethe section 2 in which the slopes could be rationals). Theorem 3.1 (i) There is an algorithm which for a tropical monotonerational function f yields its decomposition into tropical monotone binomialsand tropical monotone trinomials. The number of composants does not exceedthe number of tropical roots of f (cf. Proposition 2.1 and Remark 2.2);(ii) there is an algorithm which decomposes a tropical rational function f = g ◦ · · · ◦ g k ◦ h ◦ · · · ◦ h m ◦ g (0) k ◦ · · · ◦ g (0)1 (cf. (4)) where each g i , ≤ i ≤ k is either a tropical non-monotone rationalbinomial with ± slopes or a tropical non-monotone rational trinomial with ± slopes (cf. (1)), each h j , ≤ j ≤ m is either a tropical regular monotonebinomial or a tropical regular monotone trinomial, and each g (0) l , ≤ l ≤ k is a tropical singular monotone trinomial. The number of binomials among g , . . . , g k plus the double number of trinomials among g , . . . , g k does not exceedthe number of blocks of edges of f (cf. Theorem 2.4). The number m does notexceed the number of edges of f , and the number k equals the number of edgesof f with zero slopes (again cf. Theorem 2.4 and (3));(iii) let f be a tropical monotone rational function (respectively, a tropicalpolynomial) with the slopes of its edges a , . . . , a n ≥ (respectively, a > . . . >a n ≥ ) and denote by q i , ≤ i ≤ n the denominator of the irreducible fraction a i /a i − . Then f is a composition of tropical rational binomials (respectively,tropical binomials) iff ( q · · · q n ) | a . Remark 3.2 If f satisfies the latter condition in (iii) we call f completelydecomposable.This condition in (iii) is equivalent to a more symmetric one: for any m ≥ and j ≥ such that m + 2 j < n it holds Y ≤ i ≤ j a m +2 i | Y ≤ i ≤ j +1 a m +2 i − . In particular, for n = 2 (trinomials), the condition in (iii) for a , a , a isequivalent to a | ( a a ) . Proof . (i). If an increasing f has at least 4 edges then take any its tropicalroot x being neither the left-most nor the right-most. Define h to coincidewith f on the interval ( −∞ , x ] and g to coincide with the identity functionon the interval ( −∞ , f ( x )]. Then define h on the interval [ x , ∞ ) to coincide9ith the linear function x + f ( x ) − x , and define g on the interval [ f ( x ) , ∞ )to coincide with the composition f ◦ ( x − f ( x ) + x ). Then f = g ◦ h .Continuing in this way, applying further the described construction to g, h we complete the proof of (i).(ii). First, similar to the proof of Theorem 2.4 one represents (by means ofstraightening) f = g ◦ h (assume w.l.o.g. that the first edge of f with non-zeroslope has a positive slope), where g is either a tropical non-monotone binomialwith the slopes of its edges 1 and − , − ,
1, while h being a tropical rational function with less number of blocksof edges than f .Continuing in this way, while it is possible, we arrive to a tropical non-decreasing rational function f (0) such that f = g ◦ · · · ◦ g k ◦ f (0) (cf. (1)).Applying to f (0) the constructions from the proof of Theorem 2.4 (cf. (3)) andfrom the proof of Theorem 3.1 (i), we complete the proof of (ii).(iii). The proofs for both cases f being a tropical increasing rational func-tion or a tropical polynomial go similarly.Let f = g ◦ · · · ◦ g k where each g i , ≤ i ≤ k is a tropical increasingrational binomial (respectively, a tropical binomial) with slopes b i , c i , ≤ i ≤ k (respectively, b i > c i ). Denote by r i , ≤ i ≤ k the unique tropical root of g i . Partition R into intervals with the end-points ( g i +1 ◦ · · · ◦ g k ) − ( r i ) , ≤ i ≤ k . In case of tropical polynomials f all these end-points are the tropicalroots of f . In case of tropical increasing rational functions f all g i for which( g i +1 ◦ · · · ◦ g k ) − ( r i ) being not a tropical root of f , give a contribution into g ◦· · ·◦ g k by multiplying all the slopes of its edges on the intervals by the sameinteger, so w.l.o.g. one can assume that each ( g i +1 ◦ · · · ◦ g k ) − ( r i ) , ≤ i ≤ k is a tropical root of f .For 1 ≤ j ≤ n take the set I j of 1 ≤ i ≤ k such that ( g i +1 ◦ · · · ◦ g k ) − ( r i )is j -th root t j of f . Then a j /a j − = Q i ∈ I j ( c i /b i ). Therefore, q j | Q i ∈ I j b i . Since Q i b i = a , we conclude that ( q · · · q n ) | a .Conversely, let ( q · · · q n ) | a . Put integers b j := q j , ≤ j ≤ n − , b n := a / ( b · · · b n − ) and c j := b j a j /a j − , ≤ j ≤ n .Construct g n , . . . , g recursively. As a base of recursion take g n such thatits unique tropical root coincides with t n (observe that g n is defined uniquelyup to an additive shift, in other words, one can replace g n by g n + e, e ∈ R ).Assume that g n , . . . , g m +1 are already constructed by recursion. Then take g m such that its unique tropical root equals ( g m +1 ◦ · · · ◦ g n )( t m ). At the very laststep of recursion we adjust g by a suitable additive shift to make g ◦ · · · ◦ g n coincide with f at one (arbitrary) point. Hence f = g ◦ · · · ◦ g n . ✷ Remark 3.3
The algorithms designed in sections 2, 3 have polynomialcomplexity since after each procedure yielding a composant (cf. (1), (2), (3))either the number of blocks of edges or the number of edges drops at least byone. Tropical polynomials with commuting com-position
Let f, g be tropical polynomials without free terms (some statements belowhold also for more general tropical increasing algebraic rational functions). Inthis section we give a criterion when f ◦ g = g ◦ f . Note that the inverse f − (i. e. f ◦ f − = Id equals the identity function) is a tropical increasingalgebraic rational function. Denote by f k := f ◦ · · · ◦ f the k times iterationof f . We agree that f := Id . Note that tropical increasing algebraic rationalfunctions constitute a group with respect to the composition. Remark 4.1
Let f, g be tropical increasing algebraic rational functionsand f ◦ g = g ◦ f hold. We call x a fixed point of f if f ( x ) = x . The set F f ⊂ R of fixed points is a finite union of disjoint closed intervals { [ x i , y i ] } i (including isolated points, i. e. x i = y i ). Since f ◦ g ( x ) = g ◦ f ( x ) = g ( x ) weconclude that g ( F f ) = F f , therefore g ( x i ) = x i , g ( y i ) = y i for all i since g isincreasing.Observe that either g ( x ) = x for any point y i < x < x i +1 , either g ( x ) < x for any point y i < x < x i +1 or g ( x ) > x for any point y i < x < x i +1 . Indeed,otherwise consider the set of fixed points F g ∩ [ y i , x i +1 ] , and arguing as abovein the previous paragraph we get f ( F g ∩ [ y i , x i +1 ]) = F g ∩ [ y i , x i +1 ] , again f ( x ) = x for any end-point of an interval of F g ∩ [ y i , x i +1 ] , which contradictsthe choice of y i , x i +1 , unless F g ⊃ ( y i , x i +1 ) , in other words g ( x ) = x for anypoint y i < x < x i +1 . We allow intervals with ±∞ end-points.In the case of tropical polynomials f, g without free terms there is at mostone end-point of the intervals of fixed points of f, g , which we denote by x ,due to the convexity of f, g and taking into the account that the slopes of edgesof f, g are greater or equal than 1. Thus, there are at most two intervals ( −∞ , x ] , [ x , ∞ ) or just one interval ( −∞ , ∞ ) when x = ∞ . Theorem 4.2
Tropical polynomials f, g commute: f ◦ g = g ◦ f iff either x = ∞ and f = x + c , g = x + c , x ∈ R for some c , c ∈ R or −∞ < x < ∞ and the following is valid.There exists a tropical increasing algebraic rational function h such that h ( x ) = x (see Remark 4.1) and • either f = h p , g = h q for suitable non-negative integers p, q • or f = ax + x (1 − a ) , g = bx + x (1 − b ) for suitable integers a, b ≥ holds on the interval [ x , ∞ ) . Similarly, • either f = h k , g = h m for suitable non-negative integers k, m • or f = dx + x (1 − d ) , g = ex + x (1 − e ) for suitable integers d, e ≥ holds on the interval ( −∞ , x ] , Remark 4.3
Note that h is not necessary a tropical polynomial. roof . In one direction, namely when either such appropriate h does existor c , c do exist, obviously f ◦ g = g ◦ f holds.From now on let f ◦ g = g ◦ f . If f ( x ) = x (respectively, g ( x ) = x ) forany x ≥ x one can put h := g, f = h (respectively, h := f, g = h ) on theinterval ( x , ∞ ). Thus, from now on we suppose that f ( x ) > x, g ( x ) > x for any x > x (cf. Remark 4.1). We construct (increasing) h on the interval( −∞ , x ) and separately on the interval ( x , ∞ ) such that h ( x ) = x andafter that glue them together and obtain a tropical increasing algebraic rationalfunction h required in Theorem 4.2. Lemma 4.4
Let f, g be tropical increasing algebraic rational functions, f ◦ g = g ◦ f and for some point y i < x < x i +1 it holds f ( x ) = g ( x ) . Then f coincides with g on the interval ( y i , x i +1 ) . Proof of Lemma 4.4 . Since neither f nor g has a fixed point in theinterval ( y i , x i +1 ) one can assume for definiteness that f ( y ) > y, g ( y ) > y forany y i < y < x i +1 (see Remark 4.1). For each integer k we have f ( f k ( x )) = g ( f k ( x )). The increasing sequence x < f ( x ) < f ( x ) < · · · tends to x i +1 takinginto the account that F f ∩ ( y i x i +1 ) = ∅ . Therefore, the right-most edges of f and g on the interval ( y i , x i +1 ) coincide. Suppose that f and g do not coincideon the interval ( y i , x i +1 ).Take the left-most point z ∈ ( y i , x i +1 ) such that f ( y ) = g ( y ) for any z ≤ y < x i +1 . Hence on a sufficiently small interval [ z, z ] function f (respectively, g ) is linear ax − az + f ( z ) (respectively, bx − bz + f ( z )) and a = b . Since x i +1 > f ( z ) = g ( z ) > z and due to the choice of z there exists a linearfunction cx + d such that on a sufficiently small interval [ z, z ] the composition f ◦ g coincides with the linear function cbx − cbz + cf ( z ) + d , while thecomposition g ◦ f on [ z, z ] coincides with the linear function cax − caz + cf ( z ) + d , which contradicts to the commutativity f ◦ g = g ◦ f and provesLemma 4.4. ✷ Fix an interval ( y i , x i +1 ) for the time being and denote by T f the set oftropical roots of f . We considered the case when f ( x ) = x for any x ∈ ( y i , x i +1 )or when g ( x ) = x for any x ∈ ( y i , x i +1 ) above, so we assume that either f ( x ) > x for any x ∈ ( y i , x i +1 ) or f ( x ) < x for any x ∈ ( y i , x i +1 ), andeither g ( x ) > x for any x ∈ ( y i , x i +1 ) or g ( x ) < x for any x ∈ ( y i , x i +1 )(cf. Remark 4.1). First, we study the case ( T f ∪ T g ) ∩ ( y i , x i +1 ) = ∅ . Since g ( y i ) = f ( y i ) = y i , g ( x i +1 ) = f ( x i +1 ) = x i +1 we conclude that one or bothend-points of the interval ( y i , x x +1 ) equal ±∞ .When both y i = −∞ , x i +1 = ∞ , we have f = x + c , g = x + c for some c , c ∈ R .If y i ∈ R , x i +1 = ∞ (the case y i = −∞ , x i +1 ∈ R is analyzed in a similarway) then f (respectively, g ) coincides on the interval ( y i , ∞ ) with a linear12unction ax − ay i + y i (respectively, bx − by i + y i ) for suitable rationals a, b > T f ∪ T g ) ∩ ( y i , x i +1 ) = ∅ .From now on we again assume f, g to be tropical polynomials and let( T f ∪ T g ) ∩ ( −∞ , x ) = ∅ (cf. Remark 4.1). Lemma 4.5 (i) If h , h are tropical algebraic rational functions and x ∈ T h ◦ h then either x ∈ T h or h ( x ) ∈ T h . For tropical polynomials f, g theconverse is true: if either x ∈ T g or g ( x ) ∈ T f then x ∈ T f ◦ g ;(ii) let f ◦ g = g ◦ f . If x ∈ T g then either f − ( x ) ∈ T g or g ◦ f − ( x ) ∈ T f ;(iii) let f ◦ g = g ◦ f . If x ∈ T f \ T g then g ( x ) ∈ T f . Proof of Lemma 4.5 . (i). For the converse statement the convexity of f, g is used.(ii). Denote y := f − ( x ). Due to (i) y ∈ T g ◦ f , hence either y ∈ T g or g ( y ) ∈ T f again due to (i) and taking into the account that f ◦ g = g ◦ f ;(iii) Since due to (i) x ∈ T g ◦ f = T f ◦ g we conclude that g ( x ) ∈ T f again bymeans of (i). ✷ Consider a directed graph G with the nodes being the points from ( T f ∪ T g ) ∩ ( −∞ , x ) and the arrows according to Lemma 4.5 as follows (recall that G is not empty, the case of empty G was studied above). From every node x ∈ T g ∩ ( −∞ , x ) there is an arrow labeled by f − to the node f − ( x ), providedthat f − ( x ) ∈ T g , and there is an arrow labeled by g ◦ f − to the node g ◦ f − ( x ),provided that g ◦ f − ( x ) ∈ T f (observe that f − ( x ) , g ◦ f − ( x ) ∈ ( −∞ , x )).In addition, there is an arrow labeled by g from every node x ∈ T f \ T g to thenode g ( x ) ∈ T f (again g ( x ) ∈ ( −∞ , x )).There is a cycle in G (due to Lemma 4.5), let it contain a node x . Denoteby t the composition of the labels of the arrows (starting with x ) in this cycle.Then t ( x ) = x and one can represent t = g s ◦ f − r (taking into the accountthat f ◦ g = g ◦ f ) for some non-negative integers s, r at least one of whichbeing positive. Observe that in fact, s, r > f ( x ) < x , g ( x ) < x forany x < x (cf. Remark 4.1).Hence g s ( x ) = f r ( x ). Lemma 4.4 implies that g s coincides with f r onthe interval ( −∞ , x ). Denote n := GCD ( s, r ), then ( g s/n ◦ f − r/n ) n = Id on the interval ( −∞ , x ). The function u := g s/n ◦ f − r/n is increasing piece-wise linear. Therefore, one can partition R into a finite number of intervals(including unbounded ones) such that on each of these intervals [ y , y ] itholds u ( y ) = y , u ( y ) = y , and either u ( y ) > y for any y < y < y , either u ( y ) < y for any y < y < y or u ( y ) = y for any y < y < y (cf. Remark 4.1).Hence u = Id , i. e. g s/n = f r/n on the interval ( −∞ , x ).For appropriate positive integers i, j it holds 1 = − i ( s/n ) + j ( r/n ). Con-sider a tropical increasing algebraic rational function h := g j ◦ f − i . Then h r/n = g jr/n ◦ f − ir/n = g jr/n ◦ g − is/n = g ;13 s/n = g js/n ◦ f − is/n = f jr/n ◦ f − is/n = f on the interval ( −∞ , x ).In a similar way one produces h on the interval ( x , ∞ ), provided that x < ∞ . This completes the proof of Theorem 4.2. ✷ Remark 4.6
It would be interesting to give a criterion for commuting trop-ical increasing algebraic rational functions, and more generally, for tropicalnon-monotone algebraic rational functions.
Example 4.7
We exhibit an example of a commuting pair of increasing trop-ical rational functions f, g (defined on the interval [ x , ∞ ) , f (0) = g (0) = 0 ,thereby x = 0 , see Theorem 4.2) not satisfying the conclusion of Theorem 4.2.Pick a real < t , integers a > α > , b ≥ such that a = b, a | ( bα ) . denote u := αt, v := at, w := αat . Define g to be piece-wise linear whose graph on [ x , ∞ ) consists of three edges having the slopes a, b, a , respectively, and withthe tropical roots at the points u, w (this defines g uniquely). Similarly, define f whose graph also has three edges with the slopes α, ( bα ) /a, α , respectively,and with the tropical roots v, w . Then the graph on [ x , ∞ ) of the increasingtropical rational function f ◦ g = g ◦ f has three edges as well with the slopes αa, αb, αa , respectively, and with tropical roots t, w .In case when f k = g m (cf. Theorem 4.2) we get that α k = a m . Thus, ifthere are no such integers k, m the conclusion of Theorem 4.2 for f, g is notfulfilled. We call a polygonal line L ⊂ R n with k +1 intervals a sequence of intervals withendpoints v , . . . , v k ∈ Q n such that i -th interval has endpoints v i , v i +1 for 1 ≤ i ≤ k −
1, while unbounded 0-th interval (a ray) has v as its right endpoint, andunbounded k -th interval (a ray) has v k as its left endpoint. The vector of slopes of i -th interval, 1 ≤ i ≤ k − a i, , . . . , a i,n ) := v i +1 − v i , similarlyone can define a vector of slopes ( a , , . . . , a ,n ) of 0-th and ( a k, , . . . , a k,n ) of k -th intervals, respectively (we assume that the latter two vectors of slopes arealso rational). Note that the vector of slopes is determined up to a positivefactor.We say that tropical rational functions f , . . . , f n in one variable t providea tropical rational parametrisation of L if the map ( f , . . . , f n ) : R → L is abijection and (for definiteness) ( f , . . . , f n ) − ( v i ) < ( f , . . . , f n ) − ( v i +1 ) , ≤ i ≤ k −
1. In particular, { v , . . . , v k } coincides with the set of all the tropicalroots of f , . . . , f n , i. e. the points where one of the functions f , . . . , f n isnot smooth. We suppose w.l.o.g. that one can’t discard any v i , ≤ i ≤ while keeping the propery of L to be a polygonal line. When f , . . . , f n are tropical polynomials (respectively, Laurent polynomials), we talk about tropical polynomial (respectively, Laurent polynomial) parametrisation of L .In case if L is a subset of a tropical curve one can treat a parametrisationof L as a parametrisation of the tropical curve (cf. [3]) since a parametrisationprovides a parametric family of solutions of a system of tropical equations. Example 5.1
Let T ⊂ R be a tropical curve (a tropical line) defined by atropical polynomial min { x, y, } . Then L ⊂ T consisting of two rays { x =0 ≤ y } ∪ { y = 0 ≤ x } admits a tropical rational parametrisation with f := − min { t, } , f := − min {− t, } . Proposition 5.2
A polygonal line L has(i) always a tropical rational parametrisation;(ii) a tropical polynomial parametrisation iff a i,j ≥ , ≤ i ≤ k, ≤ j ≤ n ,and a i,j = 0 implies a l,j = 0 for all l ≥ i ;(iii) a tropical Laurent polynomial parametrisation iff • a i,j < implies a i +1 ,j < ; • a i +1 ,j > implies a i,j > ; • a i,j > , a i +1 ,j > , a i,j < , a i +1 ,j < imply a i,j /a i,j ≤ a i +1 ,j /a i +1 ,j for all ≤ i ≤ k − , ≤ j = j ≤ n . Proof . (i) We have to construct tropical univariate rational functions f , . . . , f n . First we construct piece-wise linear functions g , . . . , g n with ratio-nal slopes (in [3] such functions are called tropical Newton-Puiseux rationalfunctions). As a set of tropical roots of g , . . . , g n we take points 1 , . . . , k . Thevector of the values of g , . . . , g n at point i we put v i , ≤ i ≤ k . Thereby, g , . . . , g n are defined on interval [1 , k ] ⊂ R . To extend g , . . . , g n to interval( −∞ ,
1] (respectively, [ k, ∞ )) use the vector of the slopes of 0-th (respectively, k -th) interval of L .To proceed to tropical rational functions f , . . . , f n (so, piece-wise linearfunctions with integer slopes), denote by M the least common multiple of allthe denominators of the slopes of g , . . . , g n (i. e. the slopes of L ). As the setof tropical roots of f , . . . , f n take points 1 /M, . . . , k/M . The vector of thevalues at point i/M we put v i , ≤ i ≤ k . In other words, the correspondingslopes of f , . . . , f n are obtained from the corresponding slopes of g , . . . , g n multiplying by M . Satisfying also the latter condition, one extends f , . . . , f n to intervals ( −∞ , /M ] and [ k/M, ∞ ).(ii) If f , . . . , f n constitute a tropical polynomial parametrization of L thensince the slopes of each f j , ≤ j ≤ n (being a convex function) are non-increasing non-negative integers we get the conditions stated in (ii).15onversely, if the latter conditions are fulfilled one can recursively on i choose positive rationals c = 1 , c , . . . , c k in such a way that c i +1 · a i +1 ,j ≤ c i · a i,j , ≤ i ≤ k − , ≤ j ≤ n taking each c i +1 to be the maximal possibleamong satisfying the latter inequalities. Therefore, one can take c i · a i,j , ≤ j ≤ n as the slopes of (Newton-Puiseux polynomials [3], i. e. convex piece-wise linear functions with rational non-negative slopes) g j , ≤ j ≤ n with thetropical roots at points 1 , . . . , k . Then as at the end of the proof of (i) onecan obtain tropical polynomials f j , ≤ j ≤ n with the non-negative integerslopes M · c i · a i,j , ≤ i ≤ k, ≤ j ≤ n and with the tropical roots at points1 /M, . . . , k/M . Then f , . . . , f n provide a required parametrization of L .(iii) If there exists a tropical Laurent polynomial parametrization f , . . . , f n of L then the slopes b i,j , ≤ i ≤ k, ≤ j ≤ n of f j , ≤ j ≤ n , respectively,being integers fulfil the conditions b i,j ≥ b i +1 ,j , ≤ i ≤ k − , ≤ j ≤ n .On the other hand, there exist positive rationals c , . . . , c k such that b i,j = c i · a i,j , ≤ i ≤ k, ≤ j ≤ n . This entails the conditions from (iii).Conversely, let the conditions from (iii) be fulfilled. Construct positiverationals c = 1 , c , . . . , c k such that c i · a i,j ≥ c i +1 · a i +1 .j , ≤ i ≤ k − , ≤ j ≤ n by recursion on i . Assume that c = 1 , c , . . . , c i are already constructed.Take the maximal possible c i +1 > c i · a i,j ≥ c i +1 · a i +1 .j for all1 ≤ j ≤ n such that a i,j > , a i +1 ,j >
0. Then for suitable j for which a i,j > , a i +1 ,j > c i · a i,j = c i +1 · a i +1 ,j . For every 1 ≤ j ≤ n for which a i,j < , a i +1 ,j < a i,j /a i,j ≤ a i +1 ,j /a i +1 ,j implies c i +1 · a i +1 ,j ≤ c i · a i,j .Thus, as in (i), (ii) one first constructs piece-wise linear functions g j , ≤ j ≤ n with rational non-increasing slopes c i · a i,j , ≤ j ≤ n and with thetropical roots at points 1 , . . . , k . Denote by M the common denominator ofthese slopes and construct tropical Laurent polynomials f , . . . , f n with theslopes obtained from the slopes of g j , ≤ j ≤ n multiplying them by M and with the tropical roots 1 /M, . . . , k/M . Then f , . . . , f n provide a requiredparametrization of L . ✷ Remark 5.3
One can construct the required parametrizations in Proposi-tion 5.2 within polynomial complexity following the proofs of (i), (ii), (iii).
It would be interesting to extend parametrizations from 1-dimensionalpolygonal lines to multidimensional polyhedral complexes.
Acknowledgements . The author is grateful to the grant RSF 16-11-10075and to MCCME for inspiring atmosphere.16 eferences [1] J. von zur Gathen. Functional decomposition of polynomials: the wildcase.
J. Symbolic Comput. , 10:437–452, 1990.[2] J. von zur Gathen, J. Gutierrez and R. Rubio. Multivariate polynomialdecomposition.
Appl. Algebra Engrg. Comm. Comput. , 14:11–31, 2003.[3] D. Grigoriev. Tropical Newton-Puiseux polynomials.
Lect. Notes Comput.Sci. , 11077:177–186, 2018.[4] D. Grigoriev and V. Podolskii. Tropical combinatorial Nullstellensatz andfewnomials testing.
Lect. Notes Comput. Sci. , 10472:284–297, 2017.[5] D. Kozen, S. Landau and R. Zippel. Decomposition of algebraic functions.
J. Symbolic Comput. , 22:235–246, 1996.[6] D. Maclagan and B. Sturmfels.
Introduction to Tropical Geometry: , vol-ume 161 of
Graduate Studies in Mathematics . American MathematicalSociety, 2015.[7] G. F. Mont´ufar, R. Pascanu, K. Cho and Y. Bengio. On the number oflinear regions of deep neural networks.
Advances in Neural InformationProcessing Systems 27 , Montreal, 2924–2932, 2014.[8] F. Pakovich. Semiconjugate Rational Functions: A Dynamical Approach.
Arnold Math. J. , 4:59–68, 2018.[9] J. Ritt. Prime and composite polynomials.
Trans. Amer. Math. Soc. ,23:51–66, 1922.[10] J. Ritt. Permutable rational functions.