Supertropical Polynomials and Resultants
This paper, a continuation of , involves a closer study of polynomials of supertropical semirings and their version of tropical geometry in which we introduce the concept of relatively prime polynomials and resultants, with the aid of some topology. Polynomials in one indeterminant are seen to be relatively prime iff they do not have a common tangible root, iff their resultant is tangible. The Frobenius property yields a morphism of supertropical varieties; this leads to a supertropical version of Bézout’s theorem. Also, a supertropical variant of factorization is introduced which yields a more comprehensive version of Hilbert’s Nullstellensatz than the one given in .
Key words and phrases:Matrix algebra, Supertropical algebra, Supertropical polynomials, Relatively prime, Resultant, Bézout’s theorem.
2000 Mathematics Subject Classification:Primary 11C, 11S, 12D; Secondary 16Y60
1. Introduction and review
The supertropical algebra, a cover of the max-plus algebra, explored in , , was designed to provide a more comprehensive algebraic theory underlying tropical geometry. The abstract foundations of supertropical algebra, including polynomials over supertropical semifields, are given in . The corresponding matrix theory is explored in , and this paper is a continuation, exploring the resultant of supertropical polynomials in terms of matrices, and the ensuing applications to the resultant. The tropical resultant has already been studied by Sturmfels [5, 6], Dickenstein, Feichtner, and Sturmfels , and Tabera , but our purely algebraic approach is quite different, leading to a tropical version of Bézout’s Theorem (Theorem 5.1).
Since this paper deals mainly with polynomials and their roots, it could be viewed as a continuation of , although we explicitly state those results that we need. We briefly review the underlying notions. The underlying structure is a semiring with ghosts, which we recall is a triple where is a semiring with zero element (often identified in the examples with , as indicated below), and is a semiring ideal, called the ghost ideal, together with an idempotent semiring homomorphism
called the ghost map, i.e., which preserves multiplication as well as addition. We write for , called the -value of . Two elements and in are said to be -matched if they have the same -value; we say that dominates if . Two vectors are -matched if their corresponding entries are -matched.
Throughout this paper, we also assume the key property called supertropicality:
In particular, .
A supertropical semiring has the extra structure that is ordered, and satisfies the property called bipotence: whenever
A supertropical domain is a supertropical semiring for which is a monoid, called the set of tangible elements (denoted as when is unambiguous), such that the map (defined as the restriction from to ) is onto. We write for . We also define a supertropical semifield to be a commutative supertropical domain for which is a group; in other words, every tangible element of is invertible. Thus, is also a (multiplicative) group. Since any strictly ordered commutative semigroup has an ordered Abelian group of fractions, one can often reduce from the case of a (commutative) supertropical domain to that of a supertropical semifield.
When studying a supertropical domain , it is convenient to define an inverse function which is a retract of in the sense that is on , and writing for , we have for any (When is 1:1, we take to be on . In general, the function need not be uniquely defined if is not 1:1.)
The following natural topology is very useful in dealing with certain delicate issues.
For any supertropical domain , we define the -topology to have a base of open sets of the form
We call such sets open intervals and tangible open intervals, respectively. We say that is connected if each open interval cannot be written as the union of two nonempty disjoint intervals.
is endowed with the product topology induced by the -topology on .
Clearly is in the closure of , since any open interval containing also contains .
The -topology restricts to a topology on , whose base is the set of tangible intervals.
We often will assume that (and thus ) is divisibly closed, by passing to the divisible closure ; see [3, Section 3.4] for details.
1.1. The function semiring
Our main connection from supertropical algebra to geometry comes from supertropical functions, which we view in the following supertropical setting:
denotes the set of functions from to . A function is said to be ghost if
for every ; a function is called tangible if
for every nonempty open set of with respect to the product topology induced by Definition 1.2.
consists of the sub-semiring comprised of functions in the semiring which are continuous with respect to the -topology.
has the ghost map given by defining Thus, is a semiring with ghosts, satisfying supertropicality, although is not a supertropical semiring since bipotence fails.
The product of tangible functions is also tangible. (Indeed, by definition, for any open interval , is not ghost, so therefore is a nonempty open set. By definition, is not ghost, and thus is not ghost.)
Functions are -distinct on an open set if there is a nonempty dense open set on which for all and all .
To satisfy Definition 1.7, it is enough to find dense for each , such that for all since then one takes
The idea underlying the definition is that there is a dense subset of , at each point of which only one of the dominates.
Any polynomial can be viewed naturally in . We say that two polynomials are e-equivalent if their images in are the same; i.e., if they yield the same function from to . Abusing notation, we sometimes write for a polynomial , indicating that involves the variables .
We say that is essential in if there exists some nonempty open set for which
We define the set to be the set .
The case of an essential monomial of a polynomial, defined in , is a special case of this definition. The essential part of a polynomial is the sum of its essential monomials. Since the essential part of has the same image in as , we may assume that the polynomials we examine are essential. Note that a polynomial is ghost (as in Definition 1.4) iff its essential part is a sum of ghost monomials.
By definition, for any tangible function, there is a nonempty open set on which it cannot be ghost. Thus, a tangible essential summand of a polynomial must dominate at some tangible value.
Recall that the point is called a root of a polynomial iff is ghost. For we say that a root of is ordinary if it is a member of an open interval that does not contain any other roots of . Likewise, a common root of two polynomials and is 2-ordinary if it is a member of an open interval that does not contain any other common roots of and . (More generally, for , a root of is said to be ordinary if a belongs to some open set which contains a dense subset on which is tangible, but we only consider the case in this paper.)
Suppose is ghost on some nonempty open set on which the are -distinct. Then each summand of that is essential on is ghost on an open subset of .
Otherwise the subset of on which dominates contains a tangible element, and thus contains a tangible open set, contrary to hypothesis.∎
We say that a function is tangible at if a is not a root of , i.e., if We denote the set of these point as:
Confusion could arise because a tangible polynomial need not be tangible at every point. For example, the tangible polynomial is tangible at all tangible points except at and , where its values are ghosts. It is easy to see that a polynomial in one indeterminate is tangible iff it is tangible at all but a finite number of the tangible points. Thus, any polynomial whose essential coefficients are all tangible is tangible.
Given a polynomial we define to be a tangible polynomial according to Definition 1.4 (although need not be tangible for ). Note that .
Recall that the supertropical determinant of a matrix is defined to be the permanent, i.e., ; cf. .
2. Transformations of supertropical varieties
The root set of is the set
and is called the tangible root set of .
The tangible root set provides a tropical version of affine geometry; analogously, one would define the supertropical version of projective geometry by considering equivalence classes of tangible roots of homogeneous polynomials (where, as usual, two roots are projectively equivalent if one is a scalar multiple of the other). There is the usual way of viewing a polynomial of degree as the homogeneous polynomial , and visa versa. Since the algebra is easier to notate in the affine case, we focus on that.
We need to be able to find transformations of supertropical root sets, in order to move them away from “bad” points.
Given , we define the multiplicative translation
Clearly, when the are invertible,
Thus, the roots of are multiplicatively translated by from those of .
Given , define the additive translation
If where with “sufficiently small,” then Indeed, writing where does not appear in and dividing through by the maximal possible power of , we may assume that is nonzero, and thus dominates any root whose -th component has small enough -value. Hence is ghost, and this dominates in as well as in .
2.1. The partial Frobenius morphism
Another transformation comes from a morphism of supertropical root
sets which arises from the Frobenius property, which we
recall from [3, Remark
There is a semiring endomorphism
given by We want to refine this for polynomials.
Define the -th -Frobenius map given by
For any and , the -th -Frobenius map is a homomorphism of semirings, which is in fact an automorphism when and is a divisibly closed supertropical semifield.
Writing summed over we have
It follows just as in [3, Proposition 3.21] that for
In the set-up of Lemma 2.3, each Frobenius map defines a morphism of root sets, given by
which we call the partial Frobenius morphisms.
2.2. Supertropical Zariski topology and the generic method
Since one of the most basic tools in algebraic geometry is Zariski density, we would like to utilize the analogous tool here:
Any polynomial formula expressing equality of -values that holds on a dense subset of must hold for all of since polynomials are continuous functions.
Such a density argument is used in Section 4. There is an alternate method to Zariski density for verifying that identical relations holding for tangible polynomials must hold for arbitrary polynomials. It is not difficult to write down a generic polynomial over a semiring with tangibles and ghosts. Namely, we let where the are indeterminates over , and view the polynomial any polynomial can be obtained by specializing the accordingly. However, one has to contend with the following difficulty: Although this new semiring with ghosts satisfies supertropicality, it is not a supertropical semiring, and so identical relations holding in supertropical semirings may well fail in .
3. Supertropical polynomials in one indeterminate
This section is a direct continuation of ; we focus on properties of common tangible roots of polynomials in the supertropical setting. Assume throughout this section that is an -divisible supertropical semifield, with ghost ideal and tangible elements . We view polynomials in as functions, according to their equivalence classes in , or equivalently we consider the full polynomials [3, Definition 6.1] which are their natural representatives. Thus,
A polynomial is called monic if its leading coefficient is or (i.e., or in logarithmic notation).
We recall the following factorization:
[3, Theorem 7.43 and Corollary 7.44] Any monic full polynomial in one indeterminate has a unique factorization of the form , where the tangible component is the maximal product of tangible linear factors , and the intangible component is a product of irreducible quadratic factors of the form , at most one linear left ghost and at most one linear right ghost . (One obtains and from the factorization called “minimal in ghosts.”)
The factors of as described in the theorem are all irreducible polynomials in . Furthermore, by [3, Theorem 7.43], their sets of tangible roots are disjoint, and in fact one can read off these irreducible quadratic factors from the connected components of .
Denoting the linear tangible terms as and the quadratic terms as , we write
for this factorization of which is minimal in ghosts.
We say that a polynomial e-divides if, for a suitable polynomial , the polynomials and are e-equivalent. (A weaker concept is given below, in Definition 6.1).
Any tangible polynomial of degree has at most distinct tangible roots.
If is a tangible polynomial of degree , then e-factors uniquely into tangible linear factors.
Suppose a polynomial e-divides for tangible, and is irreducible nontangible. Then e-divides .
Write as in (3.1) where is the tangible component of . Then and is tangible; hence and must divide . ∎
We turn to the question of how to compare polynomials in terms of their roots. The next example comes as a bit of a surprise.
Some examples of polynomials such that is ghost, but and have no common tangible root.
Suppose is a full polynomial, all but one of whose monomials have a ghost coefficient, and . For example, take and Then is obviously ghost, but has no tangible roots at all; thus, and have no common tangible roots.
In logarithmic notation, where , take and Then
In particular, is tangible for all on the tangible open interval Also,
In particular, is tangible for all on the tangible open interval Thus and have no common tangible roots, although is ghost (since for all
One can complicate this example, say by taking and Nevertheless, these are the “only” kind of counterexamples, in the sense of the Proposition 3.8 below.
3.1. Graphs and roots
In this subsection, we assume that the supertropical semifield is connected, in order to apply some topological arguments.
The graph of a function is defined as the set of ordered -tuples in , where . Note that either component of could be tangible or ghost, so in a sense the graph has at most leaves.
The -graph is defined as ; i.e., we project onto the ghost values. (Note that if then .) It is more convenient to consider the tangible -graph
can be drawn in dimensions.
In this paper, we consider a polynomial in one indeterminate, so its -graph lies on a plane, and is a sequence of line segments which can change slopes only at the tangible roots of . We can describe the essential and quasi-essential monomials of as in : Writing and defining the slopes we see that the monomial is essential only if , and is quasi-essential only if . Note that when the monomial is essential (at a point ), the -graph for must change slope at . We say that a polynomial is full if each of its monomials is essential or quasi-essential.
We say that is -right (resp. left) half-tangible for if satisfies the following condition for each :
which implies for all with (resp. ).
By definition, if is -right half-tangible, all roots of must have -value and thus the tangible -graph of must have a single ray emerging from . (The analogous assertion holds for left half-tangible.)
If are polynomials without a common tangible root, with neither nor being monomials, and is ghost, then is left half-tangible and is right half-tangible (or visa versa); explicitly, there are in such that is -left half-tangible, is -right half-tangible, and for all in the tangible interval . Furthermore, in this case, (and likewise the degree of the lowest order monomial of is less than the degree of the lowest order monomial of .)
In order for to be ghost, must be ghost for each , which means that either:
is ghost of -value greater than ,
is ghost of -value greater than , or
Let (resp. ) denote the (open) set of tangible elements satisfying Condition (1) (resp. (2)). We are done unless and are disjoint, since any element of the intersection would be a common tangible root of and .
Note that must be ghost for every element in the closure of . (Indeed, if were tangible there would be some tangible interval containing for which all values of remain tangible; then, , contrary to definition of .) Likewise, is ghost for every element in the closure of .
If , then , and any tangible root of is automatically a root of . Hence, we may assume that and likewise are nonempty.
Let and . Since any cannot be a common root of and , we must have or tangible, thereby implying As noted above, is disjoint from the closure of .
Suppose is a tangible element in the boundary of (which by definition is the complement of in its closure). Then As noted above, must be ghost; if also lies in the closure of , then is also ghost, contrary to the hypothesis that and have no common tangible roots. Since is presumed connected, we must have
Write as a union of disjoint intervals, one of which we denote as . For of -value slightly more than suppose Then the slope of the tangible -graph of at must be at least as large as the slope of at , and this situation continues unless has some tangible root contrary to hypothesis. Thus, for each of -value implying for all such , and thus, by hypothesis, for all of -value .
We have also proved that is connected, and its closure is all of since otherwise has a tangible point at which the -graphs, and , both change slopes and thus must both have a tangible root. Hence, and are both tangible on the interior of .
By hypothesis, is not a monomial, and thus has some tangible root, which must have -value . The previous argument applied in the other direction (for small -values) shows that is ghost and is tangible for all of -value .
Finally, since increases faster than for it follows at once that the last assertion follows by symmetry. ∎
Conversely, if satisfy the conclusion of Proposition 3.8, then clearly are ghost. Thus, a pair of polynomials whose sum is ghost is characterized either as having a common tangible root or else satisfying the conclusion of Proposition 3.8. In particular, two polynomials of the same degree whose sum is ghost must have a common tangible root.
It is also instructive to consider the following example:
We have the following table of values for and :
Note that and is ghost, whereas they have exactly three ordinary common tangible roots, namely and ; each is also a common tangible root.
3.2. Relatively prime polynomials
In order to compare polynomials in terms of their roots, we need another notion. Given a polynomial , we write for the degree of the lowest order monomial of For example,
Two polynomials and of respective degrees and are relatively prime if there do not exist tangible polynomials and (not both ) with and such that is ghost with and .
We say that two polynomials and have a common -factor if there are polynomials with , such that e-divides and e-divides
Any monic polynomials and having a common -factor are not relatively prime. Indeed, write and and thus
(since they are -matched). On the other hand, two non-relatively prime polynomials without a common factor could be irreducible; for example for and we have is ghost, but both are irreducible, cf. Remark 3.2.
If and are both positive, then and cannot be relatively prime, since they have the common -factor . Similarly, if and then one can cancel from without affecting whether is relatively prime to . Thus, the issue of being relatively prime can be reduced to polynomials having nontrivial constant term. But then, cancelling powers of from and we may assume that and also have nontrivial constant term. Thus, so the condition that their lower degrees match is automatic.
Adjusting the leading coefficients in the definition, we may assume that and are both monic. (However, and need not be monic, as evidenced taking and in logarithmic notation; then is ghost.)
A nonconstant ghost polynomial cannot be relatively prime to any nonconstant polynomial , since or is ghost, where is any polynomial of degree with “large enough” coefficients or “small enough” coefficients respectively.
If is ghost with and then is ghost. Hence, is ghost at every point except , which implies is ghost, by continuity.
We also need the following observation to ease our computations.
Suppose the polynomial is ghost, and with Then is also ghost.
Write and where For any monomial of there is some monomial of such that is ghost. But any monomial of has the form which when added to is clearly ghost. ∎
Over a connected supertropical semifield , two non-constant monic polynomials and in are not relatively prime iff and have a common tangible root.
We may assume that and are both monic. In view of Remark 3.12, we may also assume that and are non-ghost, and have nontrivial constant term.
Suppose and are not relatively prime; i.e., is ghost for some tangible polynomials and , with and . Since we may cancel out the same power of from both and and thereby assume that and each have nontrivial constant term. We proceed as in the proof of Proposition 3.8, but with more specific attention to the tangible -graphs and , cf. Definition 3.6. We assume that and have no common tangible root. In other words, implies , for any , and likewise implies . Also, we may assume that and have no common tangible root, by Remark 3.12(4).
Let , and By hypothesis, is ghost for all But any is contained in a tangible open interval for which is tangible on so by assumption, for all and thus for all For all , it follows that and thus . Likewise, for all , we have and .
Note that as we increase the -value of a point, the slope of the graph of a polynomial can only increase; moreover, an increase of slope in the graph indicates the corresponding increase of degree of the dominant monomial at that point. We write (resp. for the maximal degree of a dominant monomial of (resp. at .
Clearly, for every in the interior of since the graphs and must have the same slope there.
By symmetry, we may assume that for small. The objective of our proof is to show that as increases, any change in the slope of arising from an increase of degree of the essential monomial of is matched by corresponding roots of , and thus (and ) — a contradiction.
We claim that the graphs and do not cross at any single tangible point (i.e. without some interval in ). Indeed, consider an arbitrary tangible point at which the graphs of and would cross, starting say with