AN ALGEBRA APPROACH TO TROPICAL MATHEMATICS

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1 Emory University: Saltman Conference AN ALGEBRA APPROACH TO TROPICAL MATHEMATICS Louis Rowen, Department of Mathematics, Bar-Ilan University Ramat-Gan 52900, Israel (Joint work with Zur Izhakian) May, 2011

2 1. Brief introduction to supertropical geometry 1. Amoebas and their degeneration For any complex affine variety W = {(z 1,..., z n ) : z i C} C (n), and any small t, define its amoeba A(W ) defined as {(log t z 1,..., log t z n ) :(z 1,..., z n ) W } (R { }) (n), graphed according to the (rescaled) coordinates log t z 1,..., log t z n. Note that log t z 1 z 2 = log t z 1 + log t z 2. Also, if z 2 = cz 1 for c << t then log t ( z 1 + z 2 ) = log t (( c + 1) z 1 ) log t z 1. The degeneration t is called the tropicalization of W, also called the tropicalization of f when W is the affine variety of a polynomial f.

3 Many invariants (dimension, intersection numbers, genus, etc.) are preserved under tropicalization and become easier to compute by passing to the tropical setting. This tropicalization procedure relies heavily on mathematical analysis, drawing on properties of logarithms. In order to bring in more algebraic techniques, and also permit generic methods, one brings in some valuation theory, following Berkovich and others.

4 2. A generic passage from (classical) affine algebraic geometry We consider t (the base of the logarithms) as an indeterminate. Define the Puiseux series of the form p(t) = τ R 0 c τ t τ, where the powers of t are taken over wellordered subsets of R, for c τ C (or any algebraically closed field of characteristic 0). For p(t) 0, define v(p(t)) := min{τ R 0 : c τ 0}. As t 0, the dominant term is c v(p(t)) t v(p(t)). The field of Puiseux series is algebraically closed, whereas v is a valuation, and Puiseux series serve as generic coefficients of polynomials describing affine varieties. We replace v by v to switch minimum to maximum.

5 3. The max-plus algebra as a bipotent semiring The max-plus algebra (with zero element adjoined) is actually a semiring. The zero element gets in the way, so we can study a semiring without zero, which we call a semiring. A semiring (R, +,, 1) is a set R equipped with two binary operations + and, called addition and multiplication, such that: 1. (R, +) is an Abelian semigroup; 2. (R,, 1 R ) is a monoid with identity element 1 R ; 3. Multiplication distributes over addition. A semiring is a semiring with a zero element 0 R satisfying a + 0 R = a, a 0 R = 0 R, a R. A semiring with negatives is a ring.

6 Given a set S and semiring R, one can define Fun(S, R) to be the set of functions from S to R, which becomes a semiring under componentwise operations.

7 FOUR NOTATIONS: Max-plus algebra: (R, +, max,, 0) Tropical notation (often used in tropical geometry): (T,,,, 0) Logarithmic notation (for examples): (T,, +,, 0) Algebraic semiring notation (for algebraic theory): (R,, +, 0, 1) We favor the algebraic semiring notation, since our point of view is algebraic.

8 Any ordered monoid M gives rise to a semiring, where multiplication is the monoid operation, and addition is taken to be the maximum. (Usually M is taken to be a group.) This semiring is bipotent in the sense that a + b {a, b}. Thus, the max-plus (tropical) algebra is viewed algebraically as a bipotent semiring. Conversely, any bipotent semiring becomes an ordered monoid, when we write a b when a + b = b.

9 4. Polynomials and matrices For any semiring R, one can define the semiring R[λ] of polynomials, namely (after adjoining 0) i N α i λ i : almost all α i = 0 R, where polynomial addition and multiplication are defined in the familiar way: ( )( ) ( α i λ i β j λ j α i β k j )λ k. j i = k i+j=k Likewise, one can define polynomials F [Λ] in a set of indeterminates Λ. Any polynomial f F [λ 1,..., λ n ] defines a graph in R (n+1), whose points are (a 1,..., a n, f(a 1,..., a n )).

10 The graph of a polynomial over the maxplus algebra is a sequence of straight lines, i.e., a polytope, and is closely related to the Newton polytope. Graph of λ 2 + 3λ + 4: In contrast to the classical algebraic theory, different polynomials over the maxplus algebra may have the same graph, i.e, behave as the same function. For example, λ 2 + λ + 7 and λ are the same over the max-plus algebra. There is a natural homomorphism Φ : R[λ 1,..., λ n ] Fun(R (n), R), and we view each polynomial in terms of its image in Fun(R (n), R). Likewise, one can define the matrix semiring M n (R) in the usual way.

11 5. Corner loci in tropicalizations Basic fact for any valuation v: If a i = 0, then v(a i1 ) = v(a i2 ) for suitable i 1, i 2. Suppose f = where p i K. Write ṽ(f) = n, i N (n) p i (t)λ i1 1 λ i n n. i N (n) v(p i (t))λ i1 1 λ i n The image under ṽ of any root of f (over the max-plus algebra) must be a point on which the maximal evaluation of f on its monomials is attained by at least two monomials. This is called a corner root, and the set of corner roots is called the corner locus. This brings us back to the max-plus algebra, since we are considering those monomials taking on maximal values.

12 6. Kapranov s Theorem Example 1. f = 10t 2 λ 3 + 9t 8 has the root λ a = t 2. Then ṽ(f) = 2λ has the corner root v(a) = 2. For f = (8t 5 +10t 2 )λ 3 +(3t+6)λ 2 +(7t 11 +9t 8 ) again ṽ(f) = 2λ 3 + 0λ 2 + 8, which as a function equals 2λ 3 +8 and again has the corner root 2. One can lift this to a root of f by building up Puiseux series with lowest term t 2, using valuation-theoretic methods. Theorem 1 (Kapranov). The tropicalization of the zero set of f coincides with the corner locus of the tropical function.

13 Kapranov s theorem leads us to evaluate polynomials on the max-plus algebra. Tropical polynomials are also viewed as piecewise linear functions f : R (n) R; then the corner locus is the domain of non-differentiability of the graph of f. Example 2. The polynomial 2x 3 + 6x + 7 over the max-plus algebra has corner locus {1, 2} since = 2 6 = 8, 6 1 = 7. Its graph (rewritten in classical algebra) consists of the horizontal line y = 7 up to x = 1, at which point it switches to the line segment y = x + 6 until x = 2, and then to the line y = 3x + 2.

14 7. Nice properties of bipotent semiring s Any bipotent algebra satisfies the amazing Frobenius property: ( ai ) m = a m i (1) for any natural number m. Any polynomial in one indeterminate can be factored by inspection, according to its roots. For example, λ 4 +4λ 3 +6λ 2 +5λ+3 has corner locus { 2, 1, 2, 4} and factors as (λ + 4)(λ + 2)(λ + ( 1))(λ + ( 2)).

15 8. Poor properties of bipotent semiring s Unfortunately, bipotent semiring s have two significant drawbacks: Bipotence does not reflect the true nature of a valuation v. If v(a) v(b) then v(a+b) {v(a), v(b)}, so bipotence holds in this situation, but if v(a) = v(b) we do not know much about v(a + b). For example, the lowest terms in two Puiseux series may or may not cancel when we take their sum. Distinct cosets of ideals need not be disjoint. In fact for any ideal I, given a, b R, if we take c I large enough, then a + c = c = b + c (a + I) (b + I). This complicates everything involving homomorphisms and factor structures. One

16 does not describe homomorphisms via kernels, but rather via congruences, which is much more complicated. Thus, the literature concerning the structure of max-plus semiring s is limited. There are remarkable theorems, but they are largely combinatoric in nature, and often the statements are hampered by the lack of a proper language. The objective of this research is to provide the language (and basic results) for a framework of the structure theory.

17 9. The supertropical semiring The structure is improved by considering a cover of our given ordered Abelian monoid, which we denote as R, rather than view R directly as a bipotent semiring. Namely, we take a monoid surjection ν : R 1 R. (Often ν is an isomorphism.) We write a ν for ν(a), for each a R 1. The disjoint union R := R 1 R becomes a multiplicative monoid under the given monoid operations on R 1 and R, when we define ab ν, a ν b both to be a ν b ν R. We extend ν to the ghost map ν : R R by taking ν to be the identity on R. Thus, ν is a monoid projection. We make R into a semiring by defining a + b = a for a ν > b ν ; b for a ν < b ν ; a ν for a ν = b ν. R so defined is called a supertropical domain.

18 Another way to view multiplication is to define ν i : R 1 R i (for i {1, } by ν 1 = 1 R1 and ν = ν. Then multiplication is given by ν i (a)ν j (b) = ν ij (ab). R is a semiring ideal of R. R 1 is called respectively the tangible submonoid of R, and R is ghost ideal also denoted as G. We could formally adjoin a zero element in a new component R 0, since this has properties both of tangible and ghost. Special Case: A supertropical semifield is a supertropical domain for which R 1 is an Abelian group.

19 Examples R 1 = (R, +), R = (R, +), and ν is the identity map (Izhakian s original example); R 1 = F (F a field), R is an ordered group, and ν : F R is a valuation. Note that we forget the original addition on the field F!

20 The innovation: The ghost ideal R is to be treated much the same way that one would customary treat the 0 element in commutative algebra. Towards this end, we write a = b if a = b or a = b + ghost. G (Accordingly, write a = 0 if a is a ghost.) Note that for a tangible, a = b iff a = b. G G This partial order =, called ghost surpasses, G is of fundamental importance in the supertropical theory, replacing equality in many analogs of theorems from commutative algebra.

21 Supertropical domains also satisfy the Frobenius property (a + b) m = a m + b m, m. A suggestive way of viewing (1) is to note that for any m there is a semiring endomorphism R R given by f f m, reminiscent of the Frobenius automorphism in classical algebra. But here the Frobenius property holds for every m. This plays an important role in our theory, and is called characteristic 1 in the literature.

22 2. Semirings with ghosts To handle polynomials and matrices directly, we need to describe the structure slightly more generally. A semiring with ghosts (R, G, ν) consists of a semiring R, a distinguished ideal called the ghost ideal G, and a semiring homomorphism ν : R G such that ν 2 = ν. If (R, G, ν) is a semiring with ghosts, then for any set S, Fun(S, R) is also a semiring with ghosts, whose ghost ideal is Fun(S, G). Bipotency fails for polynomials: (2λ + 1) + (λ + 2) = 2λ + 2.

23 If a polynomial f 0, then f cannot have any zeroes in the classical sense! But here is an alternate definition. An n-tuple a = (a 1,..., a n ) R (n) is called a root of a polynomial f R[λ 1,..., λ n ] if f(a) = 0, i.e., if f(a) G. G There are two kinds of values of f = h j, where h j are monomials: Case I At least two of the h j (a) ν are maximal (and thus equal), in this case f(a) = h j (a) ν G. Tangible roots in this case are just the corner roots. Case II There is a unique j for which h j (a) ν is maximal in G. Then f(a) = h j (a); this will be ghost when the coefficient of h j is ghost.

24 Example 3. (The tropical line) The tangible roots in D(R)[λ] of the polynomial f = λ 1 + λ are: (0, a) for a < 0; (a, 0) for a < 0; (a, a) for a > 0. The curve of tangible roots of f is comprised of three rays, all emanating from (0, 0).

25 10. The tropical version of the algebraic closure A semiring R is divisibly closed if m a R for each a R. There is a standard construction to embed a semifield into a divisibly closed, supertropical semifield. Example: The divisible closure of the maxplus semifield Z is Q, which is closed under taking roots of polynomials. Theorem 1. If two polynomials agree on an extension of a divisibly closed supertropical semifield R, then they already agree on R. The proof is an application of Farkas theorem from linear inequalities!

26 11. Factorization An example of an irreducible quadratic polynomial: λ ν λ + 7. (although λ ν λ + 7 = (λ + 2)(la + 5). But this is the only kind of example: Theorem 2. Any polynomial over a divisibly closed semifield is the product (as a function) of linear polynomials and quadratic polynomials of the form λ 2 +a ν λ+b, where b a < a.

27 Unique factorization can fail, even with respect to equivalence as functions. In one indeterminate: λ ν λ ν λ ν λ + 3 = (λ ν λ + 2)(λ ν λ + 1) = (λ ν λ + 2)(λ + ( 1))(λ + 2) = (λ ν λ + 3)(λ ν λ + 0). A geometrical interpretation of these factorizations: The tangible root set of f is the interval [ 2, 4], where 1 and 2 also are corner roots. The tangible root set of any irreducible quadratic factor λ 2 + a ν λ + b is the closed interval [ a b, a], and the union of these segments must correspond to the root set of f. Different decompositions of the tangible root set yield different factorizations. The decomposition for the factorization (λ ν λ + 2)(λ + ( 1))(λ + 2) is [ 2, 4] { 1} {2},

28 which is the decomposition best matching the geometric intuition. The decompositions of the root set for the other factorizations are respectively: [ 2, 4] [ 1, 2]; [ 1, 4] [ 2, 2]

29 In two indeterminates, we have a worse situation: (0 + λ 1 + λ 2 )(λ 1 + λ 2 + λ 1 λ 2 ) = λ 1 + λ 2 + λ λ2 2 + ν(λ 1 λ 2 ) + λ 2 1 λ 2 + λ 2 2 λ 1 = (0 + λ 1 )(0 + λ 2 )(λ 1 + λ 2 ). But this is an instance of the important phenomenon that a tropical variety can decompose in several ways as the union of irreducible varieties (in this case, either as a tropical line together with a tropical conic, or three rays).

30 3. Supertropical matrix theory Since 1 is not available in tropical mathematics, our main tool in linear algebra is the permanent A, which can be defined for any matrix A over any commutative semiring. Although the permanent is not multiplicative in general, it is multiplicative in the supertropical theory, in a certain sense, and enables us to formulate many basic notions from classical matrix theory. Assume R = (R, G, ν) is a commutative supertropical domain. Define the tropical determinant as (a i,j ) = π S n a π(1),1 a π(n),n. (2)

31 This notion is not very useful over the usual max-plus semiring : ( ) 0 0 Example 4. A = (over the max-plus 1 2 ( ) semiring Z). A = 2, but A =, so 3 4 A 2 = 5 4 = A 2. We can understand this better in the supertropical set-up. Definition 1. A matrix A is nonsingular if A is tangible; A is singular when A G 0. Example 5. In Example 4, A 2 with A 2 = 5 ν. is singular

32 Theorem 3. For any n n matrices over a supertropical semiring R, we have AB = A B. G In particular, AB = A B whenever AB is nonsingular.

33 Definition 2. The minor A i,j is obtained by deleting the i row and j column of A. The adjoint matrix adj(a) is the transpose of the matrix (a A i,j ), where a i,j =. i,j Some easy calculations: A = n j=1 a i,j a i,j, i. n j=1 a i,j a k,j = 0, k i. G nj=1 a j,i a j,k = 0, G adj(ab) = adj(b) adj(a). G adj(a t ) = adj(a) t.

34 Theorem A adj(a) = A n. 2. adj(a) = A n 1. The proof of equality (rather than just ghost surpasses) follows as a direct consequence of the celebrated theorem of Birkhoff and Von Neumann, which states that every positive doubly stochastic n n matrix is a convex combination of at most n 2 cyclic covers.

35 Definition 3. A quasi-identity is a multiplicatively idempotent matrix of tropical determinant 1, equal to the identity on the diagonal and ghost off the diagonal. Theorem 5. For any nonsingular matrix A over a supertropical semifield F, A adj(a) = A I A, for a suitable quasi-identity matrix I A. Likewise adj(a)a = A I A, for a suitable quasi-identity matrix I A. (I adj(a) = I A.) The adjoint also is used to solve the matrix equations Ax = v for tangible vectors x, v. G

36 The supertropical version of the Hamilton- Cayley theorem: The matrix A satisfies the polynomial f R[λ] if f(a) = (0); i.e., f(a) M n (G). G Theorem 6. Any matrix A satisfies its characteristic polynomial f A = λi + A. Example 6. The characteristic polynomial f A of ( ) 4 0 A = 0 1 over F = D(R), is (λ + 4)(λ + 1) + 0 = (λ+4)(λ+1), and indeed the vector (4, 0) is a eigenvector of A, with eigenvalue 4. However, there is no eigenvector having eigenvalue 1.

37 Definition 4. A vector v is a supertropical eigenvector of A, with supertropical eigenvalue β T, if Av = βv G for some m; the minimal such m is called the multiplicity of the eigenvalue (and also of the eigenvector). In Example 6, (0, 4) is a supertropical eigenvector of A having eigenvalue 1, although it is not an eigenvector. Theorem 7. The roots of the polynomial f A are precisely the supertropical eigenvalues of A.

38 4. Tropical dependence of vectors Definition 5. A subset W R (n) is tropically dependent if there is a finite sum αi w i G (n) 0, with each α i tangible; otherwise W is called tropically independent. Here is our hardest theorem: Theorem 8. Suppose R is a supertropical domain. The following three numbers are equal for a matrix: The maximum number of tropically independent rows; The maximum number of tropically independent columns; The maximum size of a square nonsingular submatrix of A.

39 Surprise: Even when the characteristic polynomial factors into n distinct linear factors, the corresponding n eigenvectors need not be supertropically independent! Example 7. A = (3) 9 The characteristic polynomial of A is f A = λ λ λ λ + 28, whose roots are 10, 9, 8, 1, which are the eigenvalues of A. The four supertropical eigenvectors comprise the matrix V = , which is singular, having determinant 112 ν.

40 This difficulty is resolved by passing to asymptotics, i.e., high enough powers of A. In contrast to the classical case, a power of a nonsingular n n matrix can be singular (and even ghost).

41 5. The resultant We now have all the tools at our disposal to define the supertropical resultant, in terms of Sylvester matrices, which enables us to determine when two polynomials have a common root and yields an algebraic proof of a version of Bezout s theorem.

42 6. Layered structure In order to handle multiple roots and derivatives, we need to consider multiple ghost layers, sorted over an ordered semiring L all of whose elements are presumed positive or 0.

43 Our main objectives: Introduce the refined layered structure and develop its basic properties, in analogy with the supertropical theory developed previously. This includes a description of polynomials and their behavior as functions. Indicate how the refined structure extends the scope of the supertropical theory, as well as the max-plus theory. For example, we can treat multiple roots by means of layers. Show how certain supertropical proofs actually become more natural in this theory. Relate these various concepts to notions already existing in the tropical literature.

44 The familiar max-plus algebra is recovered by taking L = {1}, whereas the standard supertropical structure is obtained when L = {1, }. Other useful choices of L include {1, 2, }, N, Q >0, R >0, Q, and R. The 1- layer is a multiplicative monoid corresponding to the tangible elements in the standard supertropical theory, and the l-layers for l > 1 correspond to the ghosts in the standard supertropical theory.

45 Unique factorization fails in the standard supertropical theory. Taking L = N yields enough refinement to permit us to utilize some tools of mathematical analysis. Taking L = Q >0 permits one to factor polynomials in one indeterminate into primary factors, and almost restores unique factorization in one indeterminate. The sticky point here is polynomials with a single root, which we call primary; these have the form i a i λ d i. (Unique factorization in several indeterminates still fails in certain situations, but because certain tropical hypersurfaces can be decomposed non-uniquely).

46 Our main construction: Suppose we are given a cancellative ordered monoid M. For any semiring L we define the semiring R(L, M) to be set-theoretically L M, where for k, l L, and a, b M, we define multiplication componentwise, i.e., (k, a) (l, b) = (kl, ab), (4) and addition from the rules: (k, a) + (l, b) = (k, a) if a > b, (l, b) if a < b, (k+l, a) if a = b. (5) Here, L measures the ghost level. Adding two elements of the value in M raises the ghost level accordingly. We write [k] a for (k, a), and define the map s : R L by s( [k] a ) = k, for any a M, k L. R := R(L, G) is a semiring.

47 12. Layered varieties Here are the main geometric definitions. Definition 6. An element a S is a corner root of a polynomial f if f(a) h(a) for each monomial h of f. (In other words, we need at least two monomials to attain the appropriate ghost level of f(a).) The corner locus Z corn (I) of I R[Λ] is the set of simultaneous corner roots of the functions in I. Any such corner locus will also be called an (affine) layered variety. The (affine) coordinate semiring of a layered variety Z is R[Λ] Fun(Z, R).

48 13. The component topology Definition 7. Write f = i f i for i = (i 1,..., i n ), a sum of monomials. Define the components D f,i of f to be D f,i := {a S : f(a) = f i (a)}. We write f comp,lay I for I F [λ 1,..., λ n ], if for every essential monomial f i of f there is g I (depending on D f,i ) with f Df,i g and ϑ f (a) ϑ g (a). For I F [Λ], define f = g iff either L and f = g + h f = g, or f =ν g with h s(g)-ghost, with f s(g)-ghost, Lay I = {f I : f k = g for some g I}. L

49 Theorem 9. (The layered Nullstellensatz) Suppose L = L 1 is archimedean, and F is a 1-divisibly closed, L-layered semifield, such that F 1 is archimedean. Suppose I F [Λ], and f F [Λ]. Then f k comp,lay I for some k N iff f Lay I.

50 HAPPY BIRTHDAY, DAVID