AN ALGEBRA APPROACH TO TROPICAL MATHEMATICS Louis Rowen, - PDF document

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

§ 1. Brief introduction to supertropical ge- ometry §§ 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) coordi- nates 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 trop- icalization of W , also called the tropical- ization of f when W is the affine variety of a polynomial f .

Many invariants (dimension, intersection num- bers, genus, etc.) are preserved under tropi- calization and become easier to compute by passing to the tropical setting. This tropi- calization procedure relies heavily on math- ematical analysis, drawing on properties of logarithms. In order to bring in more al- gebraic techniques, and also permit generic methods, one brings in some valuation the- ory, following Berkovich and others.

§§ 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 c τ t τ , ∑ p ( t ) = τ ∈ R ≥ 0 where the powers of t are taken over well- ordered 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 poly- nomials describing affine varieties. We replace v by − v to switch minimum to maximum.

§§ 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 † ( R, + , · , 1) is a a semiring † . 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 ele- ment 0 R satisfying a + 0 R = a, a · 0 R = 0 R , ∀ a ∈ R. A semiring with negatives is a ring.

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

FOUR NOTATIONS: Max-plus algebra: ( R , + , max , −∞ , 0) Tropical notation (often used in tropical ge- ometry): ( 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.

Any ordered monoid M gives rise to a semiring † , where multiplication is the monoid opera- tion, and addition is taken to be the maxi- mum. (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 bipo- tent semiring † . Conversely, any bipotent semiring † becomes an ordered monoid, when we write a ≤ b when a + b = b.

§§ 4. Polynomials and matrices For any semiring † R , one can define the semiring † R [ λ ] of polynomials , namely (af- ter adjoining 0 )   α i λ i :   ∑ almost all α i = 0 R  ,  i ∈ N where polynomial addition and multiplica- tion are defined in the familiar way: ( ∑ )( ∑ ) ( ) α i λ i β j λ j λ k . ∑ ∑ = α i β k − j i + j = 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 )) .

The graph of a polynomial over the max- plus algebra is a sequence of straight lines, i.e., a polytope, and is closely related to the Newton polytope. Graph of λ 2 + 3 λ + 4 : In contrast to the classical algebraic theory, different polynomials over the max- plus algebra may have the same graph, i.e, behave as the same function. For example, λ 2 + λ + 7 and λ 2 + 7 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.

§§ 5. Corner loci in tropicalizations Basic fact for any valuation v : If ∑ a i = 0 , then v ( a i 1 ) = v ( a i 2 ) for suitable i 1 , i 2 . Suppose p i ( t ) λ i 1 1 · · · λ i n ∑ f = n , i ∈ N ( n ) where p i ∈ K . Write v ( p i ( t )) λ i 1 1 · · · λ i n ∑ ˜ v ( f ) = n . i ∈ N ( n ) The image under ˜ v of any root of f (over the max-plus algebra) must be a point on which the maximal evaluation of f on its monomi- als 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.

§§ 6. Kapranov’s Theorem Example 1. f = 10 t 2 λ 3 + 9 t 8 has the root √ 9 10 t 2 . Then λ �→ a = − 3 v ( f ) = 2 λ 3 + 8 ˜ has the corner root v ( a ) = 2 . For f = (8 t 5 +10 t 2 ) λ 3 +(3 t +6) λ 2 +(7 t 11 +9 t 8 ) again v ( f ) = 2 λ 3 + 0 λ 2 + 8 , ˜ which as a function equals 2 λ 3 +8 and again has the corner root 2 . One can lift this to a root of f by building √ 9 10 t 2 , up Puiseux series with lowest term − 3 using valuation-theoretic methods. Theorem 1 (Kapranov) . The tropicaliza- tion of the zero set of f coincides with the corner locus of the tropical function.

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

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

§§ 8. Poor properties of bipotent semiring † s Unfortunately, bipotent semiring † s have two significant drawbacks: • Bipotence does not reflect the true na- ture 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 ex- ample, 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 ho- momorphisms and factor structures. One

does not describe homomorphisms via kernels, but rather via congruences, which is much more complicated. Thus, the literature concerning the struc- ture of max-plus semiring † s is limited. There are remarkable theorems, but they are largely combinatoric in nature, and often the state- ments 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.

§§ 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 for a ν > b ν ;     b for a ν < b ν ; a + b = a ν for a ν = b ν .    R so defined is called a supertropical domain † .