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# Generalising Kapranov’s Theorem For Tropical Geometry over Hyperfields

James Maxwell  Department of Mathematics, Computational Foundry, Bay Campus,
Swansea University, Fabian Way, Swansea, SA1 8EN<EMAIL_ADDRESS>james.william.maxwell@gmail

(Date: April 2022)

###### Abstract.

Kapranov’s theorem is a foundational result in tropical geometry. It states
that the set of tropicalisations of points on a hypersurface coincides
precisely with the tropical variety of the tropicalisation of the defining
polynomial. The aim of this paper is to generalise Kapranov’s theorem,
replacing the role of a valuation,
$\mathit{trop}:K\rightarrow\mathbb{R}\cup\\{-\infty\\}$, with a more general
class of hyperfield homomorphisms, $\mathbb{H}\rightarrow\mathbb{T}$, which
satisfy a relative algebraic closure condition. The map,
$\eta:\mathbb{T}\mathbb{C}\rightarrow\mathbb{T}$,
$\eta(x):=\mathrm{log}(|x|)$, where $\mathbb{T}\mathbb{C}$ is the tropical
complex hyperfield, provides an example of such a hyperfield homomorphism.

##  1\. Introduction

Hyperfields are generalisations of fields, where the addition is allowed to be
a multivalued operation. The notion of a multivalued operation was introduced
by Marty in the mid 1930’s, who discussed hypergroups. Then, Krasner in [16]
progressed the notion to hyperrings, and more recently the connection to
tropical geometry has been discussed by Viro in [18], then by Connes and
Consani in [10]. One of the main objects studied in this paper is
$\mathbb{T}\mathbb{C}$, the tropical complex hyperfield. It was first
introduced by Viro in [19], and is described as a dequantisation of the
complex numbers. The motivation of Viro’s work was to utilise structures with
multivalued addition as an algebraic foundation for tropical geometry. There
has been substantial progress made in the last several years in the
development of the algebraic theory of hyperfields. Baker and Bowler developed
a theory of matroids over hyperfields in [4], and there has been work done by
Bowler and Lorscheid on roots and multiplicities, especially characterising
multiplicities for the Krasner, sign and tropical hyperfields, in [6]. The
work completed by Jun, in [14] and [15], is a more recent study of algebraic
geometry over hyperfields. In [15] algebraic sets over hyperfields are
introduced and connected to tropical varieties, along with a scheme-theoretic
point of view leading to a demonstration that hyperrings without zero divisors
can be realised as the hyperring of global regular functions in [15, Theorem
D]. One consequence of the multivalued addition is a necessary extension of
the notion of a root. Over hyperfields roots are defined as elements at which
the polynomial outputs a set that includes zero, rather than exactly equals
zero.

The motivation for this paper comes from Kapranov’s theorem in tropical
geometry, [17, Theorem 3.1.3], which states, under mild hypotheses, the set of
roots of the tropicalisation of a polynomial $p$ coincides with the set of
tropicalisations of roots of $p$. Concretely, every root of $\mathit{trop}(p)$
lifts to a root of $p$. The connection is that the tropicalisation map, or
valuation $\mathit{trop}:K\rightarrow\mathbb{R}\cup\\{-\infty\\}$, is
naturally a hyperfield homomorphism.

We define a property of maps between hyperfields called _relatively
algebraically closed_ (RAC). We describe this precisely in Definition 2.29,
but intuitively the RAC property states that when a univariate polynomial is
pushed forward through a hyperfield homomorphism, the roots of the polynomial
can be lifted back to roots of the original polynomial. The existence of a
hyperfield homomorphism that has this property is demonstrated by the map
$\eta=\log(|\cdot|):\mathbb{T}\mathbb{C}\rightarrow\mathbb{T}$, in Theorem
2.38, where $\mathbb{T}\mathbb{C}$ is the tropical complex hyperfield and
$\mathbb{T}:=\mathbb{R}\cup\\{-\infty\\}$ is the tropical hyperfield. In
particular, the operations on $\mathbb{T}$ are the hyperaddition $\boxplus$,
defined as $a\boxplus b=\mathrm{max}(a,b)$, unless $a=b$ then $a\boxplus
b=[-\infty,a]$. The multiplication is $\odot:=+$. This is a hyperfield
analogue of the idempotent semiring structure that has been studied in [1] and
more recently presented in both [17] and [18].

The main result of this paper is a hyperfield version of Kapranov’s theorem
for RAC hyperfield homomorphisms which map to $\mathbb{T}$.

######  Theorem 1.1.

Take a RAC hyperfield homomorphism, $f:\mathbb{H}\rightarrow\mathbb{T}$, with
induced map of polynomials
$f_{*}:\mathbb{H}[X_{1}\,,\dots,\,X_{n}]\rightarrow\mathbb{T}[X_{1}\,,\dots,\,X_{n}]$,
given by applying $f$ coefficient-wise. Then for all polynomials
$p\in\mathbb{H}[X_{1}\,,\dots,\,X_{n}]$,

$V(f_{*}(p))=f(V(p)).$

A natural question regarding the RAC property of hyperfield homomorphisms is:
can we give tractable sufficient conditions to guarantee that a map between
hyperfields is RAC? This question is explored in Section 4 and partially
answered, stating sufficient conditions for a specific class of hyperfield
homomorphisms to be RAC.

### Structure of the paper

In Section 2 hyperfields and hyperfield homomorphisms will be introduced and
the definitions of polynomials and their roots over hyperfields are recalled.
This will include examples of common hyperfields. The relative algebraic
closure property is defined in Section 2, along with examples, in particular
it is shown that
$\eta=\log(|\cdot|):\mathbb{T}\mathbb{C}\rightarrow\mathbb{T}$ has this
property. Then in Section 3 the RAC property is used to construct a proof of a
hyperfield version of Kapranov’s theorem for maps
$\mathbb{H}\rightarrow\mathbb{T}$. Finally, in Section 4 the characterisation
of RAC maps is discussed. In particular, the multiplicity bound and
inheritance properties are introduced and utilised to provide sufficient
conditions for a hyperfield homomorphism to be RAC.

###  1.1. Acknowledgements

The author thanks Jeffrey Giansiracusa for the support and input during the
supervision sessions, where the ideas and content for the work was developed.
The author would also like to thank Ben Smith, Jaiung Jun and the two
anonymous referees for helpful feedback on earlier drafts. This work was
carried out during the author’s PhD and was supported by Swansea University,
College of Science Bursary EPSRC DTP (EP/R51312X/1), and the author would like
to show thanks for making the research possible.

##  2\. Background

A hyperfield is a generalisation of a field, where the addition operation is
allowed to be a multivalued operation, introduced by Marty and developed into
hyperrings by Krasner [16]. Hyperfields are one class of generalisations of
rings and fields; others include tracts (see [5]) and fuzzy rings (see [11]).
There is a discussion of this range of algebraic structures in [7]. The
connection between fuzzy rings and hyperrings is described in [12], where a
fully faithful functor from hyperfields to fuzzy rings with weak morphisms is
constructed. One particular motivation for working with hyperfields is that
the tropical semiring, where a foundational theory has been thoroughly
developed, with [17] as a general reference, has a natural analogue as the
tropical hyperfield. This section will recall the definitions and notation of
hyperfields.

###  2.1. Hyperfields

Given a set $\mathbb{H}$, then a map $\mathbb{H}\times\mathbb{H}\to
P(\mathbb{H})^{*}$ is a hyperoperation of $\mathbb{H}$ which is denoted by
$\boxplus$. Where $P(\mathbb{H})^{*}$ is the power set of $\mathbb{H}$,
explicitly, $P(\mathbb{H})^{*}$ is the set of all nonempty subsets of
$\mathbb{H}$. Then for subsets $A,B\subseteq\mathbb{H}$,

$A\boxplus B:=\bigcup_{a\in A\,,\,b\in B}(a\boxplus b).$

This definition can be extended for a string of elements. Let
$x_{1},...\,,x_{k}\in\mathbb{H}$ then we define their sum as follows;

$x_{1}\boxplus x_{2}\boxplus...\,\boxplus x_{k}=\bigcup_{x^{\prime}\in
x_{2}\boxplus...\boxplus x_{k}}x_{1}\boxplus x^{\prime}\qquad.$

The hyperoperation $\boxplus$ is called commutative and associative if it
satisfies

(2.1) $x\boxplus y=y\boxplus x$

and

(2.2) $(x\boxplus y)\boxplus z=x\boxplus(y\boxplus z)$

respectively.

######  Definition 2.3.

A canonical hypergroup is a tuple $(\mathbb{H},\boxplus,{\vmathbb 0})$, where
$\boxplus$ is a commutative and associative hyperoperation on $\mathbb{H}$
such that:

  * •

(H0) ${\vmathbb 0}\boxplus x=\\{x\\},\quad\forall\,x\in\mathbb{H}$.

  * •

(H1) For every $x\in\mathbb{H}$ there is a unique element of $\mathbb{H}$,
denoted $-x$, such that ${\vmathbb 0}\in x\boxplus-x$.

  * •

(H2) $x\in y\boxplus z$ iff $z\in x\boxplus(-y)$. This is normally referred to
as $\mathit{reversibility}$.

The reversibility condition is not required for non-canonical hypergroups, but
throughout this work only canonical hypergroups will be used so the label
canonical is dropped.

######  Definition 2.4.

A hyperring is a tuple $(\mathbb{H},\odot,\boxplus,{\vmathbb 1},{\vmathbb 0})$
such that:

  * •

$(\mathbb{H},\odot,{\vmathbb 1})$ is a commutative monoid.

  * •

$(\mathbb{H},\boxplus,{\vmathbb 0})$ is a commutative hypergroup.

  * •

(Absorption rule) ${\vmathbb 0}\odot x=x\odot{\vmathbb 0}={\vmathbb 0}$ for
all $x\in\mathbb{H}$.

  * •

(Distributive law) $a\odot(x\boxplus y)=(a\odot x)\boxplus(a\odot y)$ for all
$a,x,y\in\mathbb{H}$.

######  Definition 2.5.

A hyperring $\mathbb{H}$ is called a hyperfield if ${\vmathbb 0}\neq{\vmathbb
1}$ and every non-zero element of $\mathbb{H}$ has a multiplicative inverse.

######  Remark 2.6.

From this point onward, to clarify context, when discussing results over a
field $K$ the following notation will be used: $+,\times(\text{or}\,\cdot)$
and $\sum$. Whereas, when discussing results over a hyperfield $\mathbb{H}$
the following notation will be used: $\boxplus\,,\,\odot$ and
$\operatorname*{\scalerel*{\boxplus}{\sum}}$.

###  2.2. Examples of Hyperfields

The following are some key examples of common hyperfields used in the
literature. They will be used to demonstrate the properties that are defined
in this paper.

######  Example 2.7.

The tropical hyperfield has the underlying set $\mathbb{R}\cup\\{-\infty\\}$
and it is usually denoted as $\mathbb{T}$. The multiplication on $\mathbb{T}$
is an extension of the addition on $\mathbb{R}$:

$x\odot y:=x+y\quad\text{and}\quad x\odot-\infty=x+-\infty=-\infty.$

The hyperaddition is defined as the following multivalued operation:

(2.8) $x\boxplus y=\begin{cases}\\{\mathrm{max}(x,y)\\},\quad&\text{if}\quad
x\neq y\\\ \\{z\,|\,z\leq x\\}\cup\\{-\infty\\},\quad&\text{if}\quad
x=y\end{cases}$

The additive neutral element ${\vmathbb 0}$ of $\mathbb{T}$ is $-\infty$ and
the multiplicative neutral element ${\vmathbb 1}$ of $\mathbb{T}$ is 0. The
tropical hyperfield is a hyperfield analogue of the tropical semiring
described in [17]. The tropical hyperfield can also be defined using _min_
instead of _max_ , and $\\{\infty\\}$ instead of $\\{-\infty\\}$, which yields
an isomorphic structure. For further discussions of the tropical hyperfield
and demonstrations of its usefulness, see both [19] and [18].

The next example is a key part of the results outlined in the following
sections. The main purpose it has is as a candidate to replace the valued
field in Kapranov’s theorem.

######  Example 2.9.

The tropical complex hyperfield is usually denoted $\mathbb{T}\mathbb{C}$ and
has the complex numbers $\mathbb{C}$ as its underlying set. The standard
complex multiplication is given to $\mathbb{T}\mathbb{C}$. The hyperaddition
is defined in the following way for all $z,w\in\mathbb{C}$:

$z\boxplus w=\begin{cases}\\{c\in\mathbb{C}:|c|\leq z\\},&\quad\text{if}\quad
w=-z.\\\ z,&\quad\text{if}\quad|z|>|w|.\\\ w,&\quad\text{if}\quad|w|>|z|.\\\
\text{Shortest arc connecting}\;z\;\text{and}\;w,\;\text{with
radius}\;|z|,&\quad\text{if}\,|z|=|w|,\,z\neq\pm w.\end{cases}$

Figure 1. hyperaddition in $\mathbb{T}\mathbb{C}$ \- Where the first row is
two points in $\mathbb{T}\mathbb{C}$ and the bottom row represents the outcome
of the addition in red.

With this hyperaddition $\mathbb{T}\mathbb{C}$ is a hyperfield. The additive
neutral element ${\vmathbb 0}$ of $\mathbb{T}\mathbb{C}$ is $0$ and the
multiplicative neutral element ${\vmathbb 1}$ of $\mathbb{T}\mathbb{C}$ is
$1$. This hyperfield is discussed in [3, example 9] and was introduced in [19,
ection 6] where it is described as the dequantization of the field of complex
numbers. For further intuition as to the behaviour of the hyperaddition of
$\mathbb{T}\mathbb{C}$ see Figure 1 and [19, figure 1 in section 6].

######  Example 2.10.

A field $K$ can be viewed as a hyperfield in a trivial manner, where the
hyperaddition is defined as $x\boxplus y=\\{x+y\\}$.

######  Example 2.11.

The Krasner hyperfield has the underlying set $\\{0,1\\}$, and it is usually
denoted $\mathbb{K}$. The multiplication for $\mathbb{K}$ is defined as:

$0\odot 0=0,\,1\odot 1=1,\,0\odot 1=0.$

The hyperaddition for $\mathbb{K}$ is defined as:

$0\boxplus x=x\boxplus 0=\\{x\\}\quad\text{for}\quad x=0,1$ $1\boxplus
1=\\{0,1\\}.$

The additive neutral element ${\vmathbb 0}$ of $\mathbb{K}$ is 0 and the
multiplicative neutral element ${\vmathbb 1}$ of $\mathbb{K}$ is 1.

######  Example 2.12.

The hyperfield of signs has the underlying set $\\{-1,0,1\\}$, and it is
usually denoted by $\mathbb{S}$. The multiplication is the restriction of the
usual multiplication on the real numbers and the hyperaddition defined as:

$0\boxplus 0=0,\quad 0\boxplus 1=1,\quad 0\boxplus-1=-1,$ $1\boxplus
1=1,\quad-1\boxplus-1=-1,$ $1\boxplus-1=\\{0,1,-1\\}.$

The additive neutral element ${\vmathbb 0}$ of $\mathbb{S}$ is 0 and the
multiplicative neutral element ${\vmathbb 1}$ of $\mathbb{S}$ is 1.

######  Example 2.13.

The phase hyperfield, usually denoted by $\mathbb{P}$, has the underlying set
of $S^{1}\cup\\{0\\}$, where
$S^{1}=\\{e^{i\theta}\in\mathbb{C}\,|0\leq\theta<2\pi\,\\}$ , which is the
complex unit circle union with zero. Multiplication on $\mathbb{P}$ is
inherited from $\mathbb{C}$, and the hyperaddition is defined by the rule,

$\text{if}\quad\theta_{1}=\theta_{2}+\pi,\quad\text{then}\quad
e^{i\theta_{1}}\boxplus
e^{i\theta_{2}}=\\{0,e^{i\theta_{1}},e^{i\theta_{2}}\\}.$
$\text{If}\quad\theta_{1}<\theta_{2}<\theta_{1}+\pi,\quad\text{then}\quad
e^{i\theta_{1}}\boxplus
e^{i\theta_{2}}=\\{e^{i\theta}\,|\,\theta_{1}<\theta<\theta_{2}\\}.$
$\text{If}\quad\theta_{2}<\theta_{1}<\theta_{2}+\pi,\quad\text{then}\quad
e^{i\theta_{1}}\boxplus
e^{i\theta_{2}}=\\{e^{i\theta}\,|\,\theta_{2}<\theta<\theta_{1}\\}.$

The additive neutral element ${\vmathbb 0}$ of $\mathbb{P}$ is 0 and the
multiplicative neutral element ${\vmathbb 1}$ of $\mathbb{P}$ is 1. See [4]
and [9] for alternative but equivalent definitions of the phase hyperfield.

###  2.3. Hyperfield Homomorphisms

######  Definition 2.14.

Given hypergroups $\mathbb{H}_{1}$ and $\mathbb{H}_{2}$, with respective
hyperoperations $\boxplus_{1}$ and $\boxplus_{2}$, a hypergroup homomorphism
is a map $f:\mathbb{H}_{1}\to\mathbb{H}_{2}$, such that $f({\vmathbb
0})={\vmathbb 0}$ and $f(x\boxplus_{1}y)\subseteq f(x)\boxplus_{2}f(y)$ for
all $x,y\in\mathbb{H}_{1}$.

######  Definition 2.15.

Given hyperrings $\mathbb{H}_{1}$ and $\mathbb{H}_{2}$, with respective
hyperoperations $\boxplus_{1}$ and $\boxplus_{2}$ and multiplication
$\odot_{1}$ and $\odot_{2}$, a homomorphism of hyperrings
$f:\mathbb{H}_{1}\to\mathbb{H}_{2}$ is a map such that:

  1. (1)

$f(x\boxplus_{1}y)\subseteq f(x)\boxplus_{2}f(y)$ and $f({\vmathbb
0})={\vmathbb 0}.$

  2. (2)

$f(x\odot_{1}y)=f(x)\odot_{2}f(y)$ and $f({\vmathbb 1})={\vmathbb 1}.$

I.e., this is a homomorphism of additive hypergroups and a homomorphism of
multiplicative monoids.

######  Definition 2.16.

A map $f:\mathbb{H}_{1}\rightarrow\mathbb{H}_{2}$ between hyperfields is a
hyperfield homomorphism if it is a hyperring homomorphism.

######  Example 2.17.

The following are common examples of hyperfield homomorphisms.

$f:\mathbb{H}\rightarrow\mathbb{K},\quad f(x)=\begin{cases}1,&\text{if}\quad
x\neq 0.\\\ 0,&\text{if}\quad x=0.\end{cases}$
$\mathit{ph}:\mathbb{C}\rightarrow\mathbb{P},\quad\mathit{ph}(x)=\begin{cases}\frac{x}{|x|},&\text{if}\quad
x\in\mathbb{C}\backslash\\{0\\}.\\\ 0,&\text{if}\quad x=0.\end{cases}$
$\mathit{sgn}:\mathbb{R}\rightarrow\mathbb{S},\quad\mathit{sgn}(x)=\begin{cases}1,&\text{if}\quad
x\in\mathbb{R}_{>0}.\\\ -1,&\text{if}\quad x\in\mathbb{R}_{<0}.\\\
0,&\text{if}\quad x=0.\end{cases}$

A discussion of these and other hypefield homomorphims can be found in [2].

The next example will be a description of the hyperfield homomorphism which is
the basis for the results in this paper.

######  Example 2.18.

There exists a hyperfield homomorphism from the tropical complex hyperfield to
the tropical hyperfield. The map is denoted
$\eta:\mathbb{T}\mathbb{C}\rightarrow\mathbb{T}$, and is defined as:

$\eta(z):=\mathrm{log}(|z|).$

This homomorphism has previously appeared in a slightly different guise in
[2]. Firstly, in [2, page 341] the tropical triangle hyperfield,
$\mathbb{T}\triangle:=\mathbb{R}_{\geq 0}$ is defined and the hyperfield
homomorphism $|\cdot|:\mathbb{T}\mathbb{C}\rightarrow\mathbb{T}\triangle$ is
given as the standard absolute value. This map can also be seen as the lower
right arrow of [2, figure 1]. When $\mathbb{T}\triangle$ is discussed in [2]
it is stated that logarithm map,
$\mathrm{log}:\mathbb{T}\triangle\rightarrow\mathbb{R}\cup\\{-\infty\\}$,
induces a hyperfield structure on $\mathbb{R}\cup\\{-\infty\\}$. This
hyperfield structure is precisely the tropical hyperfield, denoted in this
work as $\mathbb{T}$, which is defined in Example 2.7. The motivation for
working with the composition map $\eta(z):=\mathrm{log}(|z|)$ is that the
tropical hyperfield is the multivalued analogue of the tropical semiring, and
therefore has been studied in more detail. The tropical triangle hyperfield is
isomorphic to the tropical hyperfield under the logarithm map, but has very
few mentions in the literature. Moreover, because of the similarities between
the tropical semiring and the tropical hyperfield, it is more natural to be
discussing a generalised version of Kapranov’s theorem utilising $\mathbb{T}$
rather than $\mathbb{T}\triangle$.

###  2.4. Polynomials over Hyperfields

This section will describe how polynomials and their roots are defined over
hyperfields.

######  Definition 2.19.

The set of polynomials in $n$-variables over a hyperfield $\mathbb{H}$ will be
denoted $\mathbb{H}[X_{1}\,,\dots,\,X_{n}]$, where elements of this set are
defined as:

(2.20) $\operatorname*{\scalerel*{\boxplus}{\sum}}_{I}c_{I}\odot
X^{i_{1}}\odot\,\cdots\,\odot
X^{i_{n}}=\operatorname*{\scalerel*{\boxplus}{\sum}}_{I}c_{I}\odot\underline{X}^{I},$

where multi-index notation is used and
$I=(i_{1}\,,\dots,\,i_{n})\subset\mathbb{Z}^{n}$ and $c_{I}\in\mathbb{H}$.

######  Remark 2.21.

Note that the notation $\mathbb{H}[X_{1}\,,\dots,\,X_{n}]$ is used only to
denote the _set_ of polynomials over $\mathbb{H}$. In general there is no ring
or hyperring structure on $\mathbb{H}[X_{1}\,,\dots,\,X_{n}]$, unlike the
specialised case where $\mathbb{H}$ is a field. Even in the univariate case it
is easy to see that the set of polynomials is not a hyperring; the multivalued
nature of the addition in $\mathbb{H}$ combined with the distributivity leads
to products of polynomials also being multivalued. For example, $(aX\boxplus
b)\odot(cX\boxplus d)\subseteq acX^{2}\boxplus(ad\boxplus bc)X\boxplus bd.$
The coefficient $(ad\boxplus bc)$ is not necessarily single valued, which
shows that multiplication of polynomials is multivalued, hence
$\mathbb{H}[X_{1}\,,\dots,\,X_{n}]$ is not a hyperring. See [14, remark 4.4
and example 4.13] for an explicit example where the behaviour of
$\mathbb{H}[X_{1}\,,\dots,\,X_{n}]$ is more controllable.

One difference to note between polynomials defined over a field and those
defined over a hyperfield occurs when the polynomial is evaluated at an
element of $\mathbb{H}$. When a polynomial defined over a hyperfield is
evaluated at an element, the output, unlike over a field, can be a set of
elements. This leads to the following notion of a root of a polynomial over
$\mathbb{H}$.

######  Definition 2.22.

Let
$p=\operatorname*{\scalerel*{\boxplus}{\sum}}_{I}c_{I}\odot\underline{X}^{I}\in\mathbb{H}[\underline{X}]$.
An element $\underline{a}=(a_{1}\,,\dots,\,a_{n})$ is a _root_ of the
polynomial if ${\vmathbb 0}\in
p(\underline{a})=\operatorname*{\scalerel*{\boxplus}{\sum}}_{I}c_{I}\odot\underline{a}^{I}$.
This allows for a natural definition of the variety of
$p(X_{1}\,,\dots,\,X_{n})$ as,

$V(p):=\\{\underline{a}=(a_{1}\,,\dots,\,a_{n})\in\mathbb{H}^{n}\,|\,{\vmathbb
0}\in p(\underline{a})\\}.$

The next definition recalls the notion of the multiplicity of a root for
univariate polynomials defined over hyperfields.

######  Definition 2.23.

([6, Def. 1.5]) Let $p(X)\in\mathbb{H}[X]$, the multiplicity of an element
$a\in\mathbb{H}$ is denoted $\mathit{mult}_{a}(p)$ and defined as,

(2.24)
$\mathit{mult}_{a}(p)=\begin{cases}0,\quad\text{if}\,a\in\mathbb{H}\,\text{is
not a root.}\\\
1+\mathrm{max}\\{\mathit{mult}_{a}(q),\,\text{if}\quad\exists\,q\in\mathbb{H}[X],\,\text{s.t.}\,p\in(X\boxplus-a)\odot
q(X)\\}.\end{cases}$

Because of the multivalued nature of multiplication of polynomials with
coefficients in a hyperfield, the polynomial $q(X)$ in $\eqref{multidef}$ is
not necessarily unique. This is the motivation behind the recursive definition
of the multiplicity. See [6] for details on the original definition and
examples of the non-uniqueness.

######  Example 2.25.

  1. (1)

If $p(X)=X^{2}\boxplus X\boxplus 1\in\mathbb{K}[X]$, then $1\in V(p)$, as

$p(1)=1\boxplus 1\boxplus 1=\mathbb{K}\ni{\vmathbb 0}.$

  2. (2)

If $p(X)=X^{2}\boxplus X\boxplus-1\in\mathbb{S}[X]$, then $-1\in V(p)$, as

$p(-1)=1\boxplus-1\boxplus-1=\mathbb{S}\ni{\vmathbb 0}.$

######  Remark 2.26.

There is a complete description of roots and corresponding multiplicities for
univariate polynomials over $\mathbb{K}$, $\mathbb{S}$ and $\mathbb{T}$ in
[6], where the results are used to demonstrate proofs of Descartes’ Rule of
Signs and Newton’s Polygon Rule.

Now that polynomials have been defined, their structure can be combined with
hyperfield homomorphisms to describe an induced map of polynomials over
hyperfields.

######  Definition 2.27.

Let $f:\mathbb{H}_{1}\rightarrow\mathbb{H}_{2}$ be a hyperfield homomorphism.
This induces a map from polynomials with coefficients in $\mathbb{H}_{1}$ to
polynomials with coefficients in $\mathbb{H}_{2}$. This map is denoted
$f_{*}:\mathbb{H}_{1}[X_{1}\,,\dots,\,X_{n}]\rightarrow\mathbb{H}_{2}[X_{1}\,,\dots,\,X_{n}]$,
and is defined for
$p=\operatorname*{\scalerel*{\boxplus}{\sum}}_{I}c_{I}\odot\underline{X}^{I}\in\mathbb{H}_{1}[X_{1}\,,\dots,\,X_{n}]$
as;

$f_{*}(p)=\operatorname*{\scalerel*{\boxplus}{\sum}}_{I}f(c_{I})\odot\underline{X}^{I}\in\mathbb{H}_{2}[X_{1}\,,\dots,\,X_{n}].$

(Note: the hyperoperations are now the operations over $\mathbb{H}_{2}$, and
$f_{*}(p)$ will be called the push-forward of $p$.)

######  Example 2.28.

Take the polynomial $p=4X^{2}-5X+1\in\mathbb{R}[X]$, the hyperfield
homomorphism $\mathit{sgn}:\mathbb{R}\rightarrow\mathbb{S}$ induces the map
$\mathit{sgn}_{*}:\mathbb{R}[X]\rightarrow\mathbb{S}[X]$, which gives
$\mathit{sgn}_{*}(p)=X^{2}\boxplus-X\boxplus 1\in\mathbb{S}[X]$.

Note that roots of $p\in\mathbb{H}_{1}[X]$ push-forward to roots of
$f_{*}(p)\in\mathbb{H}_{2}[X]$. This is precisely described in Lemma 3.2. Next
the definition of a _relatively algebraically closed_ hyperfield homomorphism
is introduced.

######  Definition 2.29.

Let $f:\mathbb{H}_{1}\rightarrow\mathbb{H}_{2}$ be a hyperfield homomorphism,
with induced map $f_{*}:\mathbb{H}_{1}[X]\rightarrow\mathbb{H}_{2}[X]$. We say
that $f$ is _relatively algebraically closed_ (RAC) if for all univariate
polynomials $p\in\mathbb{H}_{1}[X]$ and every root $b\in V(f_{*}(p))$, there
exists $a\in f^{-1}(b)$ such that $a\in V(p)$.

The map $f:\mathbb{H}_{1}\rightarrow\mathbb{H}_{2}$ being RAC has the
immediate consequence that $V(f_{*}(p))\subseteq f(V(p))$ for all
$p\in\mathbb{H}_{1}[X]$. Then, combing with Lemma 3.2, $V(f_{*}(p))=f(V(p))$.

######  Example 2.30.

The RAC property is defined for all univariate polynomials over
$\mathbb{H}_{1}$, and does not require the map
$f:\mathbb{H}_{1}\rightarrow\mathbb{H}_{2}$ to be surjective. To demonstrate,
consider a transcendental field extension
$\mathbb{Q}\rightarrow\mathbb{Q}(\alpha)$. The inclusion map
$i:\mathbb{Q}\hookrightarrow\mathbb{Q}(\alpha)$ is RAC but clearly not
surjective.

######  Definition 2.31.

A hyperfield $\mathbb{H}$ is called _algebraically closed_ if every univariate
polynomial has a root in $\mathbb{H}$.

The Krasner hyperfield $\mathbb{K}$ is algebraically closed, by [6, Remark
1.11]. Thus, a hyperfield $\mathbb{H}$ is algebraically closed if and only if
the canonical map (from Example 2.17) $f:\mathbb{H}\rightarrow\mathbb{K}$ is
RAC.

The following example outlines a RAC map which is the main motivating example
for this paper. It is the map that is studied in tropical geometry, and is the
basis for the generalisation to hyperfield homomorpshisms.

######  Example 2.32.

Let $K$ be an algebraically closed field with surjective valuation,
$\mathit{trop}:K\rightarrow\mathbb{R}\cup\\{-\infty\\}$. This is RAC
hyperfield homomorphism. This is the underlying structure investigated when
discussing valuations and tropicalisation maps in relation to tropical
geometry. See [17, section 2.1 and theorem 3.1.3] for a more detailed
description.

######  Example 2.33.

Take the map $\mathit{sgn}:\mathbb{R}\rightarrow\mathbb{S}$, and
$p=X^{2}-X+1\in\mathbb{R}[X]$. Then,
$\mathit{sgn}_{*}(p)=X^{2}\boxplus-X\boxplus 1\in\mathbb{S}[X]$. The
polynomial $p$ has an empty variety, whereas
$\mathit{sgn}_{*}(p)(1)=1\boxplus-1\boxplus 1=\mathbb{S}$, so $1\in
V(\mathit{sgn}_{*}(p))$. This demonstrates that the map
$\mathbb{R}\rightarrow\mathbb{S}$ is not a RAC map.

######  Example 2.34.

Take the map $\mathit{ph}:\mathbb{C}\rightarrow\mathbb{P}$, and the polynomial
$p=X^{2}+X+1\in\mathbb{C}[X]$. It is shown in [6, Remark 1.10], that
$\mathit{ph}_{*}(p)$ has a root at each $a=e^{i\theta}$ for all
$\pi/2<\theta<3\pi/2$. Not every element of this set can be lifted and hence
$\mathit{ph}:\mathbb{C}\rightarrow\mathbb{P}$ is not a RAC map.

###  2.5. Connecting RAC to $\mathbb{T}\mathbb{C}$

The remainder of the section will be focused on demonstrating that the map
$\eta:\mathbb{T}\mathbb{C}\rightarrow\mathbb{T}$ satisfies the RAC property.

######  Proposition 2.35.

Let $p=\operatorname*{\scalerel*{\boxplus}{\sum}}_{i=0}^{n}c_{i}\odot
X^{i}\in\mathbb{T}\mathbb{C}[X]$. An element $a\in\mathbb{T}\mathbb{C}$ is a
root of $p$ if there exist $j_{1}\,,\dots,\,j_{m}\in\\{1\,,\dots,\,n\\}$ such
that

$|c_{j_{1}}a^{j_{1}}|=\dots=|c_{j_{m}}a^{j_{m}}|>|c_{i}a^{i}|\quad\forall
i\not\in\\{j_{1}\,,\dots,\,j_{m}\\}\quad\text{and}\quad-
c_{0}\in\operatorname*{\scalerel*{\boxplus}{\sum}}_{k=1}^{m}c_{j_{k}}\odot
a^{j_{k}}.$

###### Proof.

By the definition of the hyperaddition over $\mathbb{T}\mathbb{C}$ the
monomial terms with the largest absolute value contribute to the hypersum when
the polynomial $p$ is evaluated at $a$. Thus, by the hypothesis,

$\displaystyle p(a)$
$\displaystyle=\operatorname*{\scalerel*{\boxplus}{\sum}}_{i=0}^{n}c_{i}\odot
a^{i}$
$\displaystyle=\operatorname*{\scalerel*{\boxplus}{\sum}}_{k=1}^{m}c_{j_{k}}\odot
a^{j_{k}}\boxplus c_{0},$

so, ${\vmathbb 0}\in p(a)$ if and only if
$-c_{0}\in\operatorname*{\scalerel*{\boxplus}{\sum}}_{k=1}^{m}c_{j_{k}}\odot
a^{j_{k}}$. ∎

######  Proposition 2.36.

Given $p\in\mathbb{T}[X]$, an element $a\in\mathbb{T}$ is a root of $p$ if and
only if there are two or more monomial terms in $p(a)$, such that they are
equal and greater than or equal to the remaining monomial terms.

###### Proof.

Evaluating $p$ at $a\in\mathbb{T}$ and utilising the hyperaddition over
$\mathbb{T}$ yields:

$p(a)=\operatorname*{\scalerel*{\boxplus}{\sum}}_{i=0}^{n}c_{i}\odot
a^{i}=\operatorname*{\scalerel*{\boxplus}{\sum}}_{k=1}^{m}c_{j_{k}}\odot
a^{j_{k}},$

where the monomials terms $c_{j_{k}}\odot a^{j_{k}}$, indexed over $1,\dots,m$
are all equal and greater that all other monomial terms. If $m=1$ then
$p(a)=c_{j_{k}}\odot a^{j_{k}}\notowner{\vmathbb 0}$, (ignoring the case where
$p(a)$ is a monomial and $a=-\infty$). Whereas, if $m\geq 2$, then

$p(a)=c_{j_{1}}\odot a^{j_{1}}\boxplus\dots\boxplus c_{j_{1}}\odot
a^{j_{1}}\ni{\vmathbb 0}$

by definition. ∎

######  Lemma 2.37.

Given $a,b\in\mathbb{T}$, if $a>b$, then for all $\alpha\in\eta^{-1}(a)$ and
$\beta\in\eta^{-1}(b)$, it holds that $|\alpha|>|\beta|$.

###### Proof.

Taking $a,b\in\mathbb{T}$ such that $a>b$, if $\alpha\in\eta^{-1}(a)$, and
$\beta\in\eta^{-1}(b)$ then
$\mathrm{log}(|\alpha|)=a>b=\mathrm{log}(|\beta|)$. As both the logarithm and
exponential functions preserve order,
$\mathrm{log}(|\alpha|)>\mathrm{log}(|\beta|)\Rightarrow|\alpha|>|\beta|$, as
required. ∎

######  Theorem 2.38.

The hyperfield homomorphism $\eta:\mathbb{T}\mathbb{C}\rightarrow\mathbb{T}$
is RAC.

###### Proof.

A polynomial $p=\operatorname*{\scalerel*{\boxplus}{\sum}}_{i=0}^{n}c_{i}\odot
X^{i}\in\mathbb{T}\mathbb{C}[X]$ has push-forward
$q=\eta_{*}(p)=\operatorname*{\scalerel*{\boxplus}{\sum}}_{i=0}^{n}\eta(c_{i})\odot
X^{i}\in\mathbb{T}[X]$. By Proposition 2.36, $a\in V(q)$ if and only if there
are two or more monomials terms that are equal and great than or equal than
the other monomial terms in $q(a)$. For this to occur there must exist
$k_{1}\,,\dots,\,k_{m}\in\\{1\,,\dots,\,n\\}$ such that, $\eta(c_{k_{1}})\odot
a^{k_{1}}=\,\dots\,=\eta(c_{k_{m}})\odot a^{k_{m}}>\eta(c_{i})\odot a^{i}$ for
all $i\in\\{1\,,\dots,\,n\\}\backslash\\{k_{1}\,,\dots,\,k_{m}\\}$.
Explicitly, the maximum is achieved more than once in the terms
$\eta(c_{1})\odot a^{1}\,,\dots,\,\eta(c_{n})\odot a^{n}$, and this maximum is
greater than or equal to $\eta(c_{0})$.

As $m\geq 2$, this allows for $t,t^{\prime}\in\\{k_{1}\,,\dots,\,k_{m}\\}$ to
be chosen to construct a lift of $a$ as follows:

$\widetilde{a}=\Big{(}\frac{-c_{t^{\prime}}}{c_{t}}\Big{)}^{\frac{1}{t-t^{\prime}}}.$

To confirm that $\tilde{a}$ is a lift of $a$, apply the map $\eta$ to
$\tilde{a}$ to give

$\eta(\tilde{a})=\Big{(}\eta(-c_{t^{\prime}})\odot(\eta(c_{t}))^{-1}\Big{)}^{\frac{1}{t-t^{\prime}}}.$

Furthermore, observe that $\eta(c_{t})\odot a^{t}=\eta(c_{t^{\prime}})\odot
a^{t^{\prime}}$. Then,

$\displaystyle\eta(c_{t})\odot a^{t}=\eta(c_{t^{\prime}})\odot a^{t^{\prime}}$
$\displaystyle\Rightarrow
a^{t-t^{\prime}}=\eta(c_{t^{\prime}})\odot(\eta(c_{t}))^{-1}$
$\displaystyle\Rightarrow
a=\Big{(}\eta(c_{t^{\prime}})\odot(\eta(c_{t}))^{-1}\Big{)}^{\frac{1}{t-t^{\prime}}}.$

As $\eta(x)=\mathrm{log}(|x|)=\mathrm{log}(|-x|)=\eta(-x)$, this shows that
$a=\eta(\tilde{a})$ as required.

It remains to shown that $\widetilde{a}$ is a root of the original polynomial
$p$. Note that due to Lemma 2.37,

(2.39)
$|c_{k_{1}}\widetilde{a}^{k_{1}}|=\,\dots\,=|c_{k_{m}}\widetilde{a}^{k_{m}}|>|c_{i}\widetilde{a}^{i}|,\quad\text{for
all}\quad i\not\in\\{k_{1}\,,\dots,\,k_{m}\\}.$

Furthermore, observe the relationship between the monomial terms with
$t,t^{\prime}$ exponents,

$\displaystyle c_{t}\widetilde{a}^{t}\boxplus
c_{t^{\prime}}\widetilde{a}^{t^{\prime}}$
$\displaystyle=c_{t}\Big{(}\Big{(}\frac{-c_{t^{\prime}}}{c_{t}}\Big{)}^{\frac{1}{t-t^{\prime}}}\Big{)}^{t}\boxplus
c_{t^{\prime}}\Big{(}\Big{(}\frac{-c_{t^{\prime}}}{c_{t}}\Big{)}^{\frac{1}{t-t^{\prime}}}\Big{)}^{t^{\prime}}$
$\displaystyle=(-1)^{\frac{t}{t-t^{\prime}}}(c_{t})^{\frac{-t^{\prime}}{t-t^{\prime}}}(c_{t^{\prime}})^{\frac{t}{t-t^{\prime}}}\boxplus(-1)^{\frac{t^{\prime}}{t-t^{\prime}}}(c_{t})^{\frac{-t^{\prime}}{t-t^{\prime}}}(c_{t^{\prime}})^{\frac{t}{t-t^{\prime}}}$
$\displaystyle=(-1)(-1)^{\frac{t^{\prime}}{t-t^{\prime}}}(c_{t})^{\frac{-t^{\prime}}{t-t^{\prime}}}(c_{t^{\prime}})^{\frac{t}{t-t^{\prime}}}\boxplus(-1)^{\frac{t^{\prime}}{t-t^{\prime}}}(c_{t})^{\frac{-t^{\prime}}{t-t^{\prime}}}(c_{t^{\prime}})^{\frac{t}{t-t^{\prime}}}$
$\displaystyle=\\{z\in\mathbb{C}\,:\,|z|\leq R\\},\quad\text{where}\quad
R=|c_{t}\widetilde{a}^{t}|\geq|c_{0}|.$

Then,
$\operatorname*{\scalerel*{\boxplus}{\sum}}_{j=1}^{m}c_{k_{j}}\odot\widetilde{a}^{k_{j}}\ni-
c_{0}$, which combined with (2.39) and Proposition 2.35 gives that
$\tilde{a}\in V(p)$. ∎

######  Corollary 2.40.

The tropical complex hyperfield $\mathbb{T}\mathbb{C}$ is algebraically
closed.

###### Proof.

It is a classical result in tropical geometry that the tropical semiring is
algebraically closed, see for example [13]. This is trivially equivalent to
the statement that the hyperfield $\mathbb{T}$ is algebraically closed. As the
map $\eta:\mathbb{T}\mathbb{C}\rightarrow\mathbb{T}$ is RAC and surjective,
every polynomial $p\in\mathbb{T}\mathbb{C}[X]$ has a corresponding polynomial
$\eta_{*}(p)\in\mathbb{T}[X]$, which has a root and can be lifted back to a
root of $p$. Hence, every polynomial in $\mathbb{T}\mathbb{C}[X]$ has a root.
∎

######  Remark 2.41.

The results outlined in [6] regarding root multiplicities over $\mathbb{T}$
and $\mathbb{K}$ demonstrate from a hyperfield perspective why they are both
algebraically closed.

######  Example 2.42.

Take the polynomial $p=i\odot
X^{2}\boxplus\Big{(}\frac{-1+i\sqrt{3}}{2}\Big{)}\odot
X\boxplus-1\in\mathbb{T}\mathbb{C}[X]$, then the push-forward is
$\eta_{*}(p)=0\odot X^{2}\boxplus 0\odot X\boxplus 0\in\mathbb{T}[X]$. It can
be seen that $0\in V(\eta_{*}(p))$. In accordance with the proof of Theorem
2.38, take $t=2$ and $t^{\prime}=1$. This gives,

$c_{t}=c_{2}=i,\quad-
c_{t^{\prime}}={\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}-}c_{1}=\Big{(}\frac{1-i\sqrt{3}}{2}\Big{)},\quad
1/(t-t^{\prime})=1$

Taking the template for the lift,

$\widetilde{a}=\Big{(}\frac{-c_{t^{\prime}}}{c_{t}}\Big{)}^{\frac{1}{t-t^{\prime}}}=\Big{(}\frac{1-i\sqrt{3}}{2i}\Big{)}.$

Now to confirm that this is a pull back of $0$ and is a root of $p(X)$:

$\Big{|}\Big{(}\frac{1-i\sqrt{3}}{2i}\Big{)}\Big{|}=\frac{1}{2}|1-i\sqrt{3}|=1\Rightarrow
f\Big{(}\frac{1-i\sqrt{3}}{2i}\Big{)}=0.$

Finally,

$\displaystyle p(\widetilde{a})=p\Big{(}\frac{1-i\sqrt{3}}{2i}\Big{)}$
$\displaystyle=i\Big{(}\frac{1-i\sqrt{3}}{2i}\Big{)}^{2}\boxplus\Big{(}\frac{-1+i\sqrt{3}}{2}\Big{)}\Big{(}\frac{1-i\sqrt{3}}{2i}\Big{)}\boxplus-1$
$\displaystyle=\Big{(}\frac{i-\sqrt{3}}{2}\Big{)}\boxplus\Big{(}\frac{1+i\sqrt{3}}{2i}\Big{)}\boxplus-1$
$\displaystyle=\Big{(}\frac{i-\sqrt{3}}{2}\Big{)}\boxplus-\Big{(}\frac{i-\sqrt{3}}{2}\Big{)}\boxplus-1$
$\displaystyle=\\{z\in\mathbb{C}\,:\,|z|\leq 1\\}\boxplus-1\ni{\vmathbb 0}.$

This shows that $\widetilde{a}$ is a lifted root, which demonstrates the
application of the structure of the proof of Theorem 2.38.

##  3\. A Generalisation of Kapranov’s Theorem

This section will present a specific generalisation of Kapranov’s Theorem over
hyperfields. The theorem by Kapranov [17, theorem 3.1.3] is a key result in
tropical geometry, which leads to the Fundamental Theorem of tropical geometry
in [17]. Recall the statement of Kapranov’s Theorem in tropical geometry.

######  Theorem 3.1.

Given an algebraically closed field $K$ with surjective valuation denoted
$\mathit{trop}:K\rightarrow\mathbb{R}\cup\\{-\infty\\}$, for a Laurent
polynomial $p=\sum_{I\in\mathbb{Z}^{n}}c_{I}X^{I}\in K[X_{1}^{\pm
1}\,,\dots,\,X_{n}^{\pm 1}]$,

$V(\mathit{trop}(p))=\mathit{trop}(V(p)).$

(For further details, see [17, Theorem 3.1.3]).

An essential point to recognise is that replacing the valuation with an
arbitrary hyperfield homomorphism, one containment holds automatically.

######  Lemma 3.2.

Let $f:\mathbb{H}_{1}\rightarrow\mathbb{H}_{2}$ be a hyperfield homomorphism.
For
$p=\operatorname*{\scalerel*{\boxplus}{\sum}}_{I}c_{I}\odot\underline{X}^{I}\in\mathbb{H}_{1}[X_{1}\,,\dots,\,X_{n}]$,

$f(V(p))\subseteq V(f_{*}(p)).$

###### Proof.

By definition
$f_{*}(p)=\operatorname*{\scalerel*{\boxplus}{\sum}}_{I}f(c_{I})\odot\underline{X}^{I}\in\mathbb{H}_{2}[X_{1}\,,\dots,\,X_{n}]$.
Let $\underline{a}=(a_{1}\,,\dots,\,a_{n})\in\mathbb{H}^{n}_{1}$ be a root of
$p$, meaning $\underline{a}\in V(p)$ and

(3.3) ${\vmathbb 0}\in
p(\underline{a})=p(a_{1}\,,\dots,\,a_{n})=\operatorname*{\scalerel*{\boxplus}{\sum}}_{I}c_{I}\odot\underline{a}^{I}.$

The aim is to demonstrate that $f(\underline{a})\in V(f_{*}(p))$ holds.
Firstly,

$\displaystyle f_{*}(p)(f(\underline{a}))$
$\displaystyle=\operatorname*{\scalerel*{\boxplus}{\sum}}_{I}f(c_{I})\odot
f(\underline{a})^{I}$
$\displaystyle=\operatorname*{\scalerel*{\boxplus}{\sum}}_{I}f(c_{I}\odot\underline{a}^{I})$
$\displaystyle\supseteq
f\Big{(}\operatorname*{\scalerel*{\boxplus}{\sum}}_{I}c_{I}\odot\underline{a}^{I}\Big{)}$
$\displaystyle\ni f({\vmathbb 0})={\vmathbb 0}$

The above steps use the properties of a hyperfield homomorphism to give that
${\vmathbb 0}\in f(p)(f(\underline{a}))$, yielding $f(\underline{a})\in
V(f_{*}(p))$. Hence, $f(V(p))\subseteq V(f_{*}(p))$ holds. ∎

The main result of the paper is to specifically generalise Kapranov’s Theorem
to hyperfield homomorphisms $f:\mathbb{H}\rightarrow\mathbb{T}$, which satisfy
the RAC property. The following theorem demonstrates that the RAC property for
a hyperfield homomorphism $\mathbb{H}\rightarrow\mathbb{T}$ can be used to
deduce the existence of lifts of roots for polynomials in $n$-variables.

######  Theorem 3.4 (Generalised Kapranov’s Theorem).

Given a polynomial $p\in\mathbb{H}[X_{1}\,,\dots,\,X_{n}]$ and a RAC
hyperfield homomorphism $f:\mathbb{H}\rightarrow\mathbb{T}$,

$V(f_{*}(p))=f(V(p)).$

###### Proof.

The inclusion $f(V(p))\subseteq V(f_{*}(p))$ is a direct consequence of Lemma
3.2.

The inclusion in the reverse direction, $V(f_{*}(p))\subseteq f(V(p))$, is
more interesting and requires an argument. Take a point $\underline{a}\in
V(f_{*}(p))$, so ${\vmathbb 0}\in f_{*}(p)(\underline{a})$. The aim is to
demonstrate that there exists an element in $V(p)$ that pushes forward to
$\underline{a}$. This will be done by restricting to univariate polynomials
and using the property that $f:\mathbb{H}\rightarrow\mathbb{T}$ is a RAC map,
to find an appropriate lift of $\underline{a}$.

Firstly, choose lifts $\lambda_{i}\in f^{-1}(a_{i})$, where
$\underline{a}=(a_{1}\,,\dots,\,a_{n})\in V(f_{*}(p))$ and
$\underline{\lambda}=(\lambda_{1},\dots,\lambda_{n})\in\mathbb{H}^{n}$. The
map $f:\mathbb{H}\rightarrow\mathbb{T}$ can be used to define the coordinate
wise map $F:\mathbb{H}^{n}\rightarrow\mathbb{T}^{n}\,,\quad
F(x_{1},\,\dots\,,x_{n}):=(f(x_{1})\,,\dots\,,f(x_{n}))$. For any non-zero
$D=(d_{1},\dots,d_{n})\in\mathbb{Z}^{n}$ the map
$\varphi(x):=(\lambda_{1}\odot x^{d_{1}}\,,\dots\,,\lambda_{n}\odot
x^{d_{n}})$ defines an inclusion
$\varphi:\mathbb{H}^{*}\rightarrow(\mathbb{H}^{*})^{n}$, where
$\mathbb{H}^{*}$ denotes $\mathbb{H}/\\{{\vmathbb 0}\\}$. Then, if the map
$\psi:\mathbb{T}^{*}\rightarrow(\mathbb{T}^{*})^{n}$ is defined as

$\displaystyle\psi(x):$ $\displaystyle=(f(\lambda_{1})\odot
x^{d_{1}}\,,\dots\,,f(\lambda_{n})\odot x^{d_{n}})$
$\displaystyle=(f(\lambda_{1})+d_{1}x\,,\dots\,,f(\lambda_{n})+d_{n}x),$

the diagram below is commutative:

${\mathbb{H}^{*}}$${\mathbb{T}^{*}}$${(\mathbb{H}^{*})^{n}}$${(\mathbb{T}^{*})^{n}}$$f$$\varphi$$F$$\psi$

Note that $\psi(0)=F(\underline{\lambda})=\underline{a}$. The polynomial $p$
will be pulled back through $\varphi$ to a univariate polynomial, then pushed
forward through $f_{*}$ and it will be shown that this polynomial has a root
at $0$. The RAC property will be used to lift this root back. The pullback of
$p$, denoted $\varphi^{*}p$, is the univariate polynomial defined by the
expression for $p$ where $X_{i}$ is replaced with $\lambda_{i}\odot
X^{d_{i}}$. Explicitly,

$\displaystyle\varphi^{*}p$
$\displaystyle=\operatorname*{\scalerel*{\boxplus}{\sum}}_{I}c_{I}\odot(\lambda_{1}\odot
X^{d_{1}})^{i_{1}}\odot\,\dots\,\odot(\lambda_{n}\odot X^{d_{n}})^{i_{n}},$
$\displaystyle=\operatorname*{\scalerel*{\boxplus}{\sum}}_{I}c_{I}\odot\underline{\lambda}^{I}\odot
X^{D\cdot I}\in\mathbb{H}[X].$

The pullback polynomial $\varphi^{*}p$ is pushed forward to
$f_{*}(\varphi^{*}p)\in\mathbb{T}[X]$. The image of $f_{*}(\varphi^{*}p)$ is
equal to the image of $f_{*}(p)$ when restricted to $\psi$.

$\displaystyle f_{*}(p)(\psi(X))$
$\displaystyle=f_{*}(p)\Big{(}f(\lambda_{1})\odot
X^{d_{1}}\,,\dots\,,f(\lambda_{n})\odot X^{d_{n}}\Big{)}$
$\displaystyle=\operatorname*{\scalerel*{\boxplus}{\sum}}_{I}f(c_{I})\odot
f(\underline{\lambda})^{I}\odot X^{D\cdot I}$
$\displaystyle=f_{*}(\varphi^{*}p)(X).$

The next step is to show that $f_{*}(\varphi^{*}p)$ has a root at
$0\in\mathbb{T}$. This can be seen as,

$f_{*}(\varphi^{*}p)(0)=\operatorname*{\scalerel*{\boxplus}{\sum}}_{I}f(c_{I})\odot
f(\underline{\lambda})^{I}\odot 0^{D\cdot I}.$

Then, due to the arithmetic over $\mathbb{T}$,

$\displaystyle\operatorname*{\scalerel*{\boxplus}{\sum}}_{I}f(c_{I})\odot
f(\underline{\lambda})^{I}\odot 0^{D\cdot I}$
$\displaystyle=\operatorname*{\scalerel*{\boxplus}{\sum}}_{I}f(c_{I})\odot
f(\underline{\lambda})^{I}$
$\displaystyle=\operatorname*{\scalerel*{\boxplus}{\sum}}_{I}f(c_{I})\odot\underline{a}^{I}$
(3.5) $\displaystyle=f_{*}(p)(\underline{a})\ni{\vmathbb 0}.$

This shows that $0\in V(f_{*}(\varphi^{*}p))$. Then, the property that the map
$f:\mathbb{H}\rightarrow\mathbb{T}$ is a RAC homomorphism gives that there
exists an element $\widetilde{a}\in\mathbb{H}$ such that $f(\widetilde{a})=0$
and $\widetilde{a}\in V(\varphi^{*}p)$.

Furthermore, to ensure that $\tilde{a}$ can be pushed forward to a root of $p$
the tuple $D\in\mathbb{Z}^{n}$ must be chosen with the following property.
Choose $D\in\mathbb{Z}^{n}$ such that the dot products of $D$ taken with
exponent vectors of monomial terms of
$p=\operatorname*{\scalerel*{\boxplus}{\sum}}_{I}c_{I}\odot\underline{X}^{I}$,
are all distinct. Explicitly,

$D\cdot J=d_{1}\cdot j_{1}+\dots+d_{n}\cdot j_{n}\neq d_{1}\cdot
j_{1}^{\prime}+\dots+d_{n}\cdot j_{n}^{\prime}=D\cdot J^{\prime},$

for all pairs $J,J^{\prime}\in I$. The condition $D\cdot J\neq D\cdot
J^{\prime}$ can be interpreted as $D$ not lying on the hyperplane defined by
$\underline{X}\cdot(J-J^{\prime})$. Therefore, it is possible to pick such a
$D\in\mathbb{Z}^{n}$, as the number of possible pairs $J,J^{\prime}$ is finite
and $\mathbb{Z}^{n}$ can not be covered by a finite union of hyperplanes.

The pullback $\varphi^{*}p$ utilises the requirement imposed on the tuple
$D=(d_{1}\,,\dots\,,d_{n})$. If the requirement was not imposed, this would
allow multiple monomials in the restricted polynomial to have equal exponents.
This would lead to the corresponding coefficient being a hypersum, thus the
potentially detrimental possibility of the restriction becoming a set of
polynomials rather than a single polynomial, which is what is needed here.

This condition on $D$ implies that the element $\widetilde{a}$ can then be
pushed forward through $\varphi$ to give an element $\varphi(\widetilde{a})\in
V(p)$. As the diagram commutes, this shows that the
$F(\varphi(\widetilde{a}))=\psi(f(\widetilde{a}))=\psi(0)=\underline{a}$,
which is sufficient to show that for every element of $V(f_{*}(p))$ there is a
lift to an element of $V(p)$. This demonstrates that $V(f_{*}(p))\subseteq
f(V(p))$, giving the desired result.

Hence, it has been shown that both $f(V(p))\subseteq V(f_{*}(p))$ and
$V(f_{*}(p))\subseteq f(V(p))$ hold. These taken together demonstrate the
required equality, $V(f_{*}(p))=f(V(p))$. ∎

The structure of the proof, restricting to univariate polynomials to use the
RAC property is based on the argument presented in a proof of the original
theorem in tropical geometry, as seen in [8]. Therefore, this adapted proof
along with Example 2.32 gives a proof which encompasses that of the original
Kapranov’s Theorem.

In particular, since $\eta:\mathbb{T}\mathbb{C}\rightarrow\mathbb{T}$ is RAC,
the above theorem applies to it.

##  4\. Characterising RAC Maps.

This section aims to present sufficient conditions for a hyperfield
homomorphism to be a RAC map. This will not include a complete description of
the necessary conditions for a hyperfield homomorphism to be a RAC; this
remains an open topic. To begin the exploration into RAC hyperfield
homomorphisms, a definition linking multiplicities of roots to the degree of
the polynomial.

######  Definition 4.1.

A hyperfield is said to satisfy the _multiplicity bound_ if for all univariate
polynomials, $p\in\mathbb{H}[X]$,

$\sum_{a\in\mathbb{H}}\mathrm{mult}_{a}(p)\leq\mathrm{deg}(p).$

Furthermore, a hyperfield is said to satisfy the _multiplicity equality_ if
the above inequality is an equality for all univariate polynomials.

Proposition B from [6] describes a relationship between the multiplicities of
roots over a field and the multiplicities of the push-forward of these roots
over a hyperfield. It is stated below, in notation consistent with this paper.

######  Proposition 4.2.

([6, Prop. B]) Let $K$ be a field and $\mathbb{H}$ a hyperfield with
hyperfield homomorphism $f:K\rightarrow\mathbb{H}$. Let $p\in K[X]$, with
push-forward $f_{*}(p)\in\mathbb{H}[X]$.Then,

$\mathrm{mult}_{b}(f_{*}(p))\geq\sum_{a\in f^{-1}(b)}\mathrm{mult}_{a}(p)$

for all $b\in\mathbb{H}$. Moreover, if $\mathbb{H}$ satisfies the multiplicity
bound for the polynomial $f_{*}(p)$ and $p\in K[X]$ splits into linear factors
then,

$\mathrm{mult}_{b}(f_{*}(p))=\sum_{a\in f^{-1}(b)}\mathrm{mult}_{a}(p).$

######  Lemma 4.3.

Let $f:K\rightarrow\mathbb{H}$ be a hyperfield homomorphism, with $\mathbb{H}$
satisfying the multiplicity bound. If the polynomial $p\in K[X]$ splits into
linear factors, then $V(f_{*}(p))\subseteq f(V(p))$ for $p$.

###### Proof.

Take the polynomial $p\in K[X]$, $f_{*}(p)\in\mathbb{H}[X]$. Take $b\in
V(f_{*}(p))$, so ${\vmathbb 0}\in f_{*}(p)(b)$ and hence
$\mathrm{mult}_{b}(f_{*}(p))>0$. As $p$ splits into linear factors over $K$
and $\mathbb{H}$ satisfies the multiplicity bound, by Proposition 4.2 we have,

$0<\mathrm{mult}_{b}(f_{*}(p))=\sum_{a\in f^{-1}(b)}\mathrm{mult}_{a}(p).$

Therefore, there exists $\hat{a}\in f^{-1}(b)$ such that
$\mathrm{mult}_{\hat{a}}(p)>0$. Thus, $\hat{a}$ is a root of $p\in K[X]$. This
demonstrates that given an element $b\in V(f_{*}(p))$ there exists an element
$\hat{a}$, such that $f(\hat{a})=b$ and $p(\hat{a})=0$. This shows that
$V(f_{*}(p))\subseteq f(V(p))$. ∎

The above lemma provides a method for detecting whether a hyperfield
homomorphism from a field to a hyperfield is a RAC map. The next stage is to
understand whether this view can be extended to maps
$f:\mathbb{H}_{1}\rightarrow\mathbb{H}_{2}$. Proposition 4.2 is the key tool
used in the proof of Lemma 4.3. The following discussion aims to explore a
generalisation of this result for maps
$f:\mathbb{H}_{1}\rightarrow\mathbb{H}_{2}$. Firstly, it is important to
recognise that this is not a simple generalisation and the property from Prop.
B in [6] does not always hold in this less restrictive setting.

######  Example 4.4.

Take the hyperfield homomorphism $f:\mathbb{P}\rightarrow\mathbb{K}$. Note
that, over the Krasner hyperfield $\mathbb{K}$, the multiplicity bound
achieves equality for all polynomials:
$\sum_{b\in\mathbb{K}}\mathrm{mult}_{b}(p)=\mathrm{deg}(p)$ for all
$p\in\mathbb{K}[X]$, (see Remark 1.11 in [6] for details). Take the polynomial
$p=X^{2}\boxplus X\boxplus 1\in\mathbb{P}[X]$, then due to the argument
presented in [6, remark 1.10],
$\sum_{a\in\mathbb{P}}\mathrm{mult}_{a}(p)=\infty$, in particular
$a=e^{i\theta}$ is a root of $p$ for all
$\frac{\pi}{2}<\theta<\frac{3\pi}{2}$. Now the push-forward coefficients of
the polynomial are unchanged, $f_{*}(p)=X^{2}\boxplus X\boxplus 1$, with
$\sum_{b\in\mathbb{K}}\mathrm{mult}_{b}(f(p))=2$. This then leads to,

$\infty=\sum_{a\in\mathbb{P}\setminus\\{0\\}}\mathrm{mult}_{a}(p)=\sum_{a\in
f^{-1}(1)}\mathrm{mult}_{a}(p)\not\leq\mathrm{mult}_{1}(f(p))=2.$

This demonstrates that the property does not hold in total generality over all
hyperfield homomorphisms.

There are several key properties of fields that underpin the result in
Proposition 4.2. These properties do not automatically hold for hyperfields.
The first key property is that all fields satisfy the multiplicity bound. The
second has to do with the factorisation process of polynomials. Restricting to
hyperfields with the multiplicity bound is a solution to half the problem,
whereas the factorisation property needs to be discussed in further detail.

If $K$ is a field, then $K[X]$ is a unique factorization domain. Moreover, if
$K$ is algebraically closed then every polynomial $p\in K[X]$ factorises
completely into a product of linear factors. It is not obvious that these
properties extend to hyperfields. This is due to the non-uniqueness of the
choice of factorisation, even for more well-behaved hyperfields, such as those
with the doubly distributive property (see [9] for description of doubly
distributive hyperfields). It could occur that for two distinct roots, the
maximum multiplicity is achieved with different factorisations. In an attempt
to overcome this the next definition is introduced.

######  Definition 4.5.

Given a polynomial $p\in\mathbb{H}[X]$, denote the list of roots, inclusive of
repetitions corresponding to multiplicities, by
$A_{p}:=\\{a_{1}\,\dots\,a_{k}\\}$. A hyperfield $\mathbb{H}$ is said to have
the _inheritance_ property at $p$, if for all subsets
$\\{a_{j_{1}},\,\dots\,,a_{j_{m}}\\}\subseteq A_{p}$, with
$m\leq\mathrm{deg}(p)$, there exists $q\in\mathbb{H}[X]$ such that

$p\in(X-a_{j_{1}})\odot(X-a_{j_{2}})\odot\dots\odot(X-a_{j_{m}})\odot q.$

A hyperfield is said to have the _inheritance_ property in general if it holds
for every polynomial $p\in\mathbb{H}[X]$.

The work in this paper does not extend to characterising the multiplicity
bound or inheritance properties, but rather opens this area up for
exploration. The following conjectures are based on the current knowledge of
doubly distributive hyperfields, specifically including
$\mathbb{K},\mathbb{S}$ and $\mathbb{T}$.

######  Conjecture 4.6.

All hyperfields with the doubly distributive property satisfies the
multiplicity bound.

######  Conjecture 4.7.

All hyperfields with the doubly distributive property satisfies the
inheritance property.

There will now be a demonstration of the implications of the multiplicity
bound and the inheritance property.

######  Lemma 4.8.

Given the hyperfield homomorphism $f:\mathbb{H}_{1}\rightarrow\mathbb{H}_{2}$,
then for all $p=\operatorname*{\scalerel*{\boxplus}{\sum}}_{i=0}^{n}c_{i}\odot
X^{i}\in\mathbb{H}_{1}[X]$,

(4.9) $\mathrm{mult}_{b}(f_{*}(p))\geq\sum_{a\in
f^{-1}(b)}\mathrm{mult}_{a}(p)$

holds if $\mathbb{H}_{1}$ satisfies the multiplicity bound and inheritance
property.

###### Proof.

As $\mathbb{H}_{1}$ satisfies the multiplicity bound the list of roots,
including repetitions corresponding to the multiplicities,
$\\{a_{1}\,\dots\,a_{k}\\}$ is a finite set with $k\leq\mathrm{deg}(p)$. Take
the subset
$\\{a_{j_{1}}\,\dots\,a_{j_{m}}\\}\subseteq\\{a_{1}\,\dots\,a_{k}\\}$, such
that $f(a_{j_{1}})=\dots=f(a_{j_{m}})=b$. These are the only elements of
$\mathbb{H}_{1}$ that are roots of $p$ and push-forward to $b$. By the
inheritance property, there exists a $q\in\mathbb{H}_{1}[X]$, such that,

$p\in(X-a_{j_{1}})\odot\dots\odot(X-a_{j_{m}})\odot q.$

By the hyperfield homomorphism properties it can be seen that under $f_{*}$,

$\displaystyle f_{*}(p)$
$\displaystyle\in(X-f(a_{j_{1}}))\odot\dots\odot(X-f(a_{j_{m}}))\odot
f_{*}(q)$ $\displaystyle\in(X-b)\odot\dots\odot(X-b)\odot f_{*}(q).$

This gives $\mathrm{mult}_{b}(f_{*}(p))\geq m$, implying that,

$\mathrm{mult}_{b}(f_{*}(p))\geq\sum_{a\in f^{-1}(b)}\mathrm{mult}_{a}(p).$

∎

######  Theorem 4.10.

Given $\mathbb{H}_{1}$ which satisfies the multiplicity equality and has the
inheritance property, and $\mathbb{H}_{2}$ that satisfies the multiplicity
bound, then the homomorphism $f:\mathbb{H}_{1}\rightarrow\mathbb{H}_{2}$ is a
RAC map.

###### Proof.

The hyperfield $\mathbb{H}_{1}$ satisfying the multiplicity equality and the
inheritance property implies by Lemma 4.8 that,
$\mathrm{mult}_{b}(f_{*}(p))\geq\sum_{a\in f^{-1}(b)}\mathrm{mult}_{a}(p)$
holds. These together then imply that;

$\mathrm{deg}(p)=\sum_{a\in\mathbb{H}_{1}}\mathrm{mult}_{a}(p)\leq\sum_{b\in\mathbb{H}_{2}}\mathrm{mult}_{b}(f_{*}(p))\leq\mathrm{deg}(f_{*}(p))=\mathrm{deg}(p).$

Explicitly, the first equality is given by the multiplicity equality of
$\mathbb{H}_{1}$ and the second inequality holds due to the multiplicity bound
on $\mathbb{H}_{2}$. Yielding, $\mathrm{mult}_{b}(f_{*}(p))=\sum_{a\in
f^{-1}(b)}\mathrm{mult}_{a}(p)$. Then finally,
$\mathrm{mult}_{b}(f_{*}(p))=\sum_{a\in f^{-1}(b)}\mathrm{mult}_{a}(p)$ gives
that the map $f:\mathbb{H}_{1}\rightarrow\mathbb{H}_{2}$ is RAC, using
analogous logic to the proof of Lemma 4.3. ∎

######  Remark 4.11.

Theorem 4.10 demonstrates that there are sufficient conditions that can be
given to both hyperfields to give the corresponding homomorphism as RAC,
although this does not classify all RAC maps. It does incorporate the
motivating example for the the paper,
$\mathit{trop}:K\rightarrow\mathbb{R}\cup\\{-\infty\\}$. Although, it can be
seen in Example 4.12 that $\mathbb{T}\mathbb{C}$ does not satisfy the
multiplicity bound, and hence does not fulfil the conditions of Theorem 4.10.
This demonstrates the theoretical complexity in attempting to outline the
conditions for a hyperfield homomorphism to be RAC.

######  Example 4.12.

Given the polynomial $p=X^{2}\boxplus X\boxplus 1\in\mathbb{T}\mathbb{C}[X]$,
then

$\displaystyle p(-1)$ $\displaystyle=(-1)^{2}\boxplus-1\boxplus 1$
$\displaystyle=1\boxplus-1\boxplus 1$
$\displaystyle=\Big{(}\\{a\in\mathbb{C}\,:\,|a|\leq 1\\}\Big{)}\boxplus 1$
$\displaystyle\ni-1\boxplus 1\ni 0$ $\displaystyle p(i)$
$\displaystyle=-1\boxplus i\boxplus 1$ $\displaystyle=\Big{(}\\{\text{shortest
closed arc between}\,-1\,\text{and}\,i\\}\Big{)}\boxplus 1$
$\displaystyle\ni-1\boxplus 1\ni 0$ $\displaystyle p(-i)$
$\displaystyle=-1\boxplus-i\boxplus 1$ $\displaystyle=\Big{(}\\{\text{shortest
closed arc between}\,-1\,\text{and}\,-i\\}\Big{)}\boxplus 1$
$\displaystyle\ni-1\boxplus 1\ni 0$

These three calculations over $\mathbb{T}\mathbb{C}$ imply that
$\\{-1,i,-i\\}\subset V(p)\subset\mathbb{T}\mathbb{C}$. This shows that a
degree 2 polynomial can have three distinct root over $\mathbb{T}\mathbb{C}$,
thus the multiplicity bound does not hold.

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# Casimir-Lifshitz Interaction between Dielectric Heterostructures

Arash Azari, Himadri S. Samanta, and Ramin Golestanian  Department of Physics
and Astronomy, University of Sheffield, Sheffield S3 7RH, United Kingdom

###### Abstract

The interaction between arbitrary dielectric heterostructures is studied
within the framework of a recently developed dielectric contrast perturbation
theory. It is shown that periodically patterned dielectric or metallic
structures lead to oscillatory lateral Casimir-Lifshitz forces, as well as
modulations in the normal force as they are displaced with respect to one
another. The strength of these oscillatory contributions increases with
decreasing gap size and increasing contrast in the dielectric properties of
the materials used in the heterostructures.

###### pacs:

05.40.-a, 81.07.-b, 03.70.+k, 77.22.-d

##  I Introduction

In light of the ongoing miniaturization of mechanical devices and the recent
developments in Casimir-Lifshitz interactions Casimir48 ; Lifshitz ; measure ;
lateral-exp ; trench , there has been some recent interest in the effect of
these interactions between the components of small mechanical devices nanomech
. Since these interaction are particularly strong at small distances, it will
be interesting to know how they can be utilized for designing novel mechanical
systems that could work without physical contact and could potentially help
solve the wear problem machine .

In the past few years there have been a surge of interest in developing
techniques that can be used to study the Casimir-Lifshitz interaction in non-
ideal geometries, including geometry perturbation theories GK ; EHGK ;
lambrecht , semiclassical semiclass  and classical ray optics Jaffe
approximations, multiple scattering and multipole expansions balian ; klich ;
multipole1 ; multipole2 ; multipole3 , world-line method gies  and exact
numerical diagonalization methods Emig-exact ; valery , as well as the
numerical Green’s function calculation method Johnson . These methods have
been used in studying the Casimir force in a variety of different geometries,
which have improved significantly our understanding of the nontrivial geometry
dependence of this effect.

The effect of non-ideal geometry has been shown to lead to a number
interesting effects. For example, it has been suggested that corrugated
surfaces opposite one another can experience an oscillatory lateral Casimir
force GK , which was subsequently observed experimentally lateral-exp . A
recent experiment probing the normal Casimir force between a smooth surface
and surface with tall rectangular corrugations also revealed further evidence
on the non-additive nature of the Casimir force trench . Here, we study the
Casimir-Lifshitz interaction between arbitrary dielectric heterostructures
within the framework of a recently developed formalism ramin ; rg-09 . We
derive a closed form expression for the Casimir-Lifshitz energy between two
dielectric heterostructures (such as the example depicted in Fig. 1) up to the
second order in the perturbation theory and show that a coherent coupling
between the different modes of the spectrum of the dielectric pattern takes
place across the gap. As a special example, we consider unidirectional
periodic heterostructures (see Fig. 1) and calculate the lateral and normal
Casimir-Lifshitz force between them within the same order in the perturbation
theory. We find that coupling between modes with identical wavevectors of the
pattern structures between the different objects can lead to modulations in
the normal force and can give rise to oscillatory later forces, reminiscent of
the lateral Casimir force that appears due to coupling between geometrical
features such as corrugations GK ; lateral-exp .

Figure 1: Schematic representation of two identical semi-infinite and periodic
objects made of intercalated layers of high and low dielectric functions,
occupying the fractions of $f$ and $1-f$, respectively. Here $H$ is the
separation between them, $a$ is a dimensionless lateral displacement, and
$\lambda$ is the wavelength of the periodic structure.

This paper is organized as follows. Section II sketches the dielectric
contrast perturbation theory, and Sec. III elaborates on how it can be used
for dielectric heterostructures giving closed form expressions for the second
order term in the perturbation theory. Section IV gives the results for the
lateral and normal Casimir-Lifshitz force for a number of choices of
materials, and Sec. V contains some discussions and concluding remarks.

##  II Theoretical Formulation

To calculate the Casimir-Lifshitz interaction we need to quantize the
electromagnetic field in a background that includes the dielectric or metallic
objects that modify the quantum fluctuations of the field. Describing a
general assortment of dielectric and metallic objects in space via a frequency
dependent dielectric profile $\epsilon(\omega,{\bf r})$, we can write a
general expression for the Casimir-Lifshitz energy as rg-09

$E=\hbar\int_{0}^{\infty}\frac{d\zeta}{2\pi}\;{\rm tr}\ln\left[{\cal
K}_{ij}(\zeta;{\bf r},{\bf r}^{\prime})\right],$ (1)

where

${\cal K}_{ij}=\left[\frac{\zeta^{2}}{c^{2}}\epsilon(i\zeta,{\bf
r})\delta_{ij}+\partial_{i}\partial_{j}-\partial_{k}\partial_{k}\delta_{ij}\right]\delta^{3}({\bf
r}-{\bf r}^{\prime}).$ (2)

We can consider the dielectric function as $\epsilon(i\zeta,{\bf
r})=1+\delta\epsilon(i\zeta,{\bf r})$, and expand Eq. (1) in powers of the
dielectric contrast. A similar approach has been the subject of a few recent
studies barton ; ramin ; buhmann ; rudi ; milton .

The expansion leads to the decomposition of ${\cal K}_{ij}$ into a diagonal
part ${\cal K}_{0,ij}$, corresponding to the empty space, and a perturbation
part $\delta{\cal K}_{ij}$, namely

${\cal K}_{ij}(\zeta;{\bf q},{\bf q}^{\prime})={\cal K}_{0,ij}(\zeta,{\bf
q})(2\pi)^{3}\delta^{3}({\bf q}+{\bf q}^{\prime})+\delta{\cal
K}_{ij}(\zeta;{\bf q},{\bf q}^{\prime}),$ (3)

where

${\cal K}_{0,ij}(\zeta,{\bf
q})=\frac{\zeta^{2}}{c^{2}}\delta_{ij}+q^{2}\delta_{ij}-q_{i}q_{j},$ (4)

and

$\delta{\cal K}_{ij}(\zeta;{\bf q},{\bf
q}^{\prime})=\frac{\zeta^{2}}{c^{2}}\delta_{ij}\delta\tilde{\epsilon}(i\zeta,{\bf
q}+{\bf q}^{\prime}).$ (5)

This yields an expansion

${\rm tr}\ln[{\cal K}]={\rm tr}\ln[{\cal
K}_{0}]+\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n}\;{\rm tr}[({\cal
K}_{0}^{-1}\delta{\cal K})^{n}],$ (6)

where

$\displaystyle{\cal K}_{0,ij}^{-1}(\zeta,{\bf
q})=\frac{\frac{\zeta^{2}}{c^{2}}\delta_{ij}+q_{i}q_{j}}{\frac{\zeta^{2}}{c^{2}}\left[\frac{\zeta^{2}}{c^{2}}+q^{2}\right]}.$
(7)

The first term is the vacuum energy in the absence of the objects, and the
terms in the series take account of their effect in a perturbative scheme. The
$n$-th order term in Eq. (6) takes on the explicit form

${\rm tr}[({\cal K}_{0}^{-1}\delta{\cal K})^{n}]=\int\frac{d^{3}{\bf
q}^{(1)}}{(2\pi)^{3}}\cdots\frac{d^{3}{\bf
q}^{(n)}}{(2\pi)^{3}}\frac{[\frac{\zeta^{2}}{c^{2}}\delta_{i_{1}i_{2}}+q_{i_{1}}^{(1)}q_{i_{2}}^{(1)}]\cdots[\frac{\zeta^{2}}{c^{2}}\delta_{i_{n}i_{1}}+q_{i_{n}}^{(n)}q_{i_{1}}^{(n)}]}{[\frac{\zeta^{2}}{c^{2}}+q^{(1)2}]\cdots[\frac{\zeta^{2}}{c^{2}}+q^{(n)2}]}\delta\tilde{\epsilon}(i\zeta,-{\bf
q}^{(1)}+{\bf q}^{(2)})\cdots\delta\tilde{\epsilon}(i\zeta,-{\bf q}^{(n)}+{\bf
q}^{(1)}),$ (8)

which involves the Fourier transform of the dielectric contrast profile. Going
to real space, we can rewrite the energy of the system as rg-09

$E=\hbar\int_{0}^{\infty}\frac{d\zeta}{2\pi}\;\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n}\int
d^{3}{\bf r}_{1}\cdots d^{3}{\bf r}_{n}{\cal A}_{i_{1}i_{2}}({\bf r}_{1}-{\bf
r}_{2})\cdots{\cal A}_{i_{n}i_{1}}({\bf r}_{n}-{\bf
r}_{1})\left[\frac{\delta\epsilon(i\zeta,{\bf
r}_{1})}{1+\frac{1}{3}\delta\epsilon(i\zeta,{\bf
r}_{1})}\right]\cdots\left[\frac{\delta\epsilon(i\zeta,{\bf
r}_{n})}{1+\frac{1}{3}\delta\epsilon(i\zeta,{\bf r}_{n})}\right],$ (9)

where

${\cal A}_{ij}(\zeta,{\bf r})=\frac{\zeta^{2}}{c^{2}}\frac{{\rm e}^{-\zeta
r/c}}{4\pi r}\left[\delta_{ij}\left(1+\frac{c}{\zeta
r}+\frac{c^{2}}{\zeta^{2}r^{2}}\right)-\frac{r_{i}r_{j}}{r^{2}}\left(1+3\frac{c}{\zeta
r}+3\frac{c^{2}}{\zeta^{2}r^{2}}\right)\right],$ (10)

We now use this formulation to study the Casimir-Lifshitz interaction between
structures with inhomogeneous or patterned dielectric properties.

##  III Dielectric Heterostructures

Let us now consider a configuration similar to the one depicted in Fig. 1,
namely two dielectric heterostructures that are placed parallel to each other
at a separation $H$. Using the definition ${\bf r}=({\bf x},z)$, the
dielectric profile can be written as

$\epsilon(i\zeta,{\bf r})=\left\\{\begin{array}[]{ll}\epsilon_{u}(i\zeta,{\bf
x}),&\;\;\;\;\frac{H}{2}\leq z<+\infty,\\\ \\\
1,&\;\;\;\;\frac{-H}{2}<z<\frac{H}{2},\\\ \\\ \epsilon_{d}(i\zeta,{\bf
x}),&\;\;\;\;-\infty<z\leq\frac{-H}{2},\end{array}\right.$ (11)

using the labels u and d for the “up” and “down” bodies respectively.

To keep the calculations tractable, we now focus on the second order term in
the series expansion in Eq. (9). For such two semi-infinite bodies, the second
order interaction term between the bodies can be written as

$E_{2}=-\frac{\hbar}{2\pi^{2}c^{2}}\int_{0}^{\infty}d\zeta\;\zeta^{2}\int
d^{2}{\bf x}d^{2}{\bf x}^{\prime}\int\frac{d^{2}{\bf Q}}{(2\pi)^{2}}\;{\rm
e}^{i{\bf Q}\cdot({\bf x}-{\bf x}^{\prime})}\;{\cal
E}(Q)\left[\frac{\delta\epsilon_{u}(i\zeta,{\bf
x})}{1+\frac{1}{3}\delta\epsilon_{u}(i\zeta,{\bf
x})}\right]\left[\frac{\delta\epsilon_{d}(i\zeta,{\bf
x}^{\prime})}{1+\frac{1}{3}\delta\epsilon_{d}(i\zeta,{\bf
x}^{\prime})}\right],$ (12)

for any lateral dielectric function profile, where

${\cal
E}(Q)=\int_{1}^{\infty}dp\;\frac{[2p^{4}-2p^{2}+1]}{\left[4p^{2}+(cQ/\zeta)^{2}\right]^{3/2}}\;{\rm
e}^{-\frac{\zeta H}{c}\sqrt{4p^{2}+(cQ/\zeta)^{2}}},$ (13)

and $\delta\epsilon_{u,d}(i\zeta,{\bf x})=\epsilon_{u,d}(i\zeta,{\bf x})-1$.

We now focus on the specific example of unidirectional periodic structures as
depicted in Fig. 1, which is made of subsequent layers of materials with
relatively high and low dielectric functions. We can use the periodic
properties of the dielectrics and write them in Fourier series expansion. As
Fig. 1 shows, we can define the dielectric profile of the $d$-object as

$\epsilon_{d}\left(i\zeta,x\right)=\left\\{\begin{array}[]{ll}\epsilon_{l}\left(i\zeta\right),&\;\;\;\;-\frac{\lambda}{2}+s\lambda\leq
x\leq-\frac{f\lambda}{2}+s\lambda,\\\ \\\
\epsilon_{h}\left(i\zeta\right),&\;\;\;\;-\frac{f\lambda}{2}+s\lambda<x<\frac{f\lambda}{2}+s\lambda,\\\
\\\ \epsilon_{l}\left(i\zeta\right),&\;\;\;\;\frac{f\lambda}{2}+s\lambda\leq
x\leq\frac{\lambda}{2}+s\lambda,\end{array}\right.$ (14)

where $s$ is an integer number. We define the Fourier series as

$\frac{\delta\epsilon_{d}\left(i\zeta,x\right)}{1+\frac{1}{3}\delta\epsilon_{d}\left(i\zeta,x\right)}=\sum_{m=-\infty}^{\infty}{\mathcal{C}}_{m}(i\zeta)\;{\rm
e}^{i2\pi mx/\lambda},$ (15)

where

${\mathcal{C}}_{m}(i\zeta)=\frac{\sin m\pi
f}{m\pi}\left[\frac{\delta\epsilon_{h}\left(i\zeta\right)}{1+\frac{1}{3}\delta\epsilon_{h}\left(i\zeta\right)}-\frac{\delta\epsilon_{l}\left(i\zeta\right)}{1+\frac{1}{3}\delta\epsilon_{l}\left(i\zeta\right)}\right],$
(16)

for $m\neq 0$, and

${\mathcal{C}}_{0}(i\zeta)=f\left[\frac{\delta\epsilon_{h}\left(i\zeta\right)}{1+\frac{1}{3}\delta\epsilon_{h}\left(i\zeta\right)}\right]+(1-f)\left[\frac{\delta\epsilon_{l}\left(i\zeta\right)}{1+\frac{1}{3}\delta\epsilon_{l}\left(i\zeta\right)}\right].$
(17)

We can find the corresponding expansion for the $u$-object by changing
$x\rightarrow x+a\lambda$.

Using the Fourier series expansion, one can find the Casimir-Lifshitz energy
between two dielectric heterostructures as depicted in Fig. 1 [up to second
order in the Clausius-Mossotti expansion of Eq. (9)] as

$E_{pp}=-\frac{\hbar
A}{2\pi^{2}c^{2}}{\sum_{m=0}^{\infty}}^{{}^{\prime}}\int_{0}^{\infty}d\zeta\;\zeta^{2}{\cal
E}\left(\frac{2\pi m}{\lambda}\right)\;{\mathcal{C}}^{2}_{m}(i\zeta)\cos(2\pi
ma),$ (18)

where the prime on the summation sign indicates that the $m=0$ term is counted
with half the weight, and the $pp$ index means the energy calculated for the
plate-plate geometry. This result shows that similar to the case of two
corrugated surfaces, two patterned dielectric heterostructures also couple to
each other at the leading order when the two wavelengths of the modulations
are equal GK . Moreover, higher harmonics contribute to the Casimir-Lifshitz
energy with exponentially decaying contributions, such that at large
separations only the fundamental mode (lowest harmonic) will survive Emig-
exact .

##  IV The Normal and Lateral Forces

We now use Eq. (18) to calculate the normal and lateral forces between
different types of dielectric and metallic heterostructures. We look at three
different types of materials as examples, namely, gold, silicon, and
air/vacuum, and consider layered materials made of gold-silicon, silicon-air,
and gold-air. We describe the dielectric function of gold using a plasma
model, namely,
$\displaystyle\epsilon(i\zeta)=1+\frac{\omega_{p}^{2}}{\zeta^{2}},$ where
$\omega_{p}$ is the plasma frequency, which is given as $\omega_{p}({\rm
Au})=1.37\times 10^{16}$ rad/s Palik . For silicon we use the Drude-Lorentz
form
$\displaystyle\epsilon(i\zeta)=1+\frac{\omega_{p}^{2}}{\zeta^{2}+\omega_{0}^{2}},$
where $\omega_{p}({\rm Si})=3.3\;\omega_{0}({\rm Si})$ and $\omega_{0}({\rm
Si})=6.6\times 10^{15}$ rad/s Palik . Finally, for air/vacuum we use
$\epsilon(i\zeta)=1$.

Due to difficulties in keeping the surfaces of the objects parallel to each
other, most experiments are performed in plate-sphere geometry. To perform the
calculation of the forces for the plate-sphere configuration, we can use the
Derjaguin Approximation Israelachvili92 , where we replace one of the semi-
infinite objects with a planar surface with a sphere with radius $R$. The
approximation is valid provided that the radius of sphere is much larger than
the distance between the dielectric heterostructures, namely, $R\gg H$. Using
this approximation we can find the normal force between a semi-infinite
dielectric heterostructure and a sphere of the same material composition as
Israelachvili92

$F_{ps}^{nor}=2\pi R\left(\frac{E_{pp}}{A}\right).$ (19)

Using this result, we can find the Casimir-Lifshitz energy for plate-sphere
configuration as

$E_{ps}=-\int_{H}^{\infty}dH^{\prime}\;F_{ps}^{nor}(H^{\prime}),$ (20)

which we can now use to calculate the lateral Casimir force as

$F_{ps}^{lat}=-\frac{1}{\lambda}\;\frac{\partial E_{ps}}{\partial a}.$ (21)

Substituting Eqs. (19) and (20) into Eq. (21), it reads

$F_{ps}^{lat}=\frac{2\pi R}{\lambda}\;\frac{\partial}{\partial
a}\int_{H}^{\infty}dH^{\prime}\;\left(\frac{E_{pp}(H^{\prime})}{A}\right).$
(22)

The above equations are the basis of the results that will be presented below.

Figure 2: Normal Casimir-Lifshitz force between layered dielectric
heterostructures as shown in Fig. 1 in the plate-sphere geometry,
corresponding to (a) gold-silicon, (b) silicon-air, and (c) gold-air, with
$f=0.5$, and to (d) gold-silicon, (e) silicon-air, and (f) gold-air, with
$f=0.2$. The numerical value of corrugation wavelength used is $\lambda=1$
$\mu$m. Different curves correspond to different gap sizes of $H=100$ nm,
$H=300$ nm, and $H=600$ nm. The forces are normalized to $F_{ps}^{0}$ that
corresponds to the normal force when the laterally-averaged dielectric profile
is used.

Figures 2a-c show the normal Casimir-Lifshitz force between two unidirectional
(layered) dielectric heterostructures as shown in Fig. 1 when laterally
displaced with respect to one another by $a\lambda$. It corresponds to the
symmetric case with $f=0.5$, and the corrugation wavelength of $\lambda=1$
$\mu$m. Three different compositions of gold-silicon, silicon-air, and gold-
air are considered each at three different gap sizes of $H=100$ nm, $H=300$
nm, and $H=600$ nm. The normal forces are normalized using the normal force
$F_{ps}^{0}$ that corresponds to the Casimir-Lifshitz force calculated within
the same scheme but with laterally averaged dielectric profile, which
corresponds to the $m=0$ term in the expansion in Eq. (18). The normal force
is found to oscillate as a function of the lateral displacement, having the
maximum value when the regions of high dielectric constant from both sides are
exactly opposite one another, and the minimum value when in the staggered
configuration where regions of higher dielectric constant face regions of
lower dielectric constant. The amplitude of the oscillations increases by
decreasing the gap size, and the effect is progressively stronger when the
contrast between the dielectric properties of the two regions is more
pronounced, with a maximum relative change of 0.7 % for gold-silicon, 7 % for
silicon-air, and 65 % for gold-air, at the closest separation of $H=100$ nm.

In Figs. 2d-f the normal Casimir-Lifshitz forces between the same types of
structures as above are presented, for the asymmetric case of $f=0.2$. One can
see two noticeable differences with the symmetric case. First, the
oscillations are now asymmetric, as enforced by the asymmetry of the
dielectric profile, although the asymmetry weakens as the gaps size increases
and eventually disappears—i.e. the oscillations become symmetric and
harmonic—at sufficiently large separations. This is consistent with the
picture that different harmonics of the dielectric contrast profile in Eq.
(18) couple with each other via an exponential terms that decays with the
corresponding wavelengths of each harmonic and as a result any asymmetry
caused by higher harmonics will die out at large gap sizes. The second new
feature is the significant enhancement of the amplitude of the oscillatory
behavior as a function of the lateral displacement. While it is still the case
that this amplitude increases with increasing contrast between the dielectric
properties of the two materials used in the layered structure, the maximum
relative change is 0.4 % for gold-silicon, 6 % for silicon-air, and 200 % for
gold-air, at the closest separation of $H=100$ nm.

Figure 3: Lateral Casimir-Lifshitz force between layered dielectric
heterostructures as shown in Fig. 1 in the plate-sphere geometry,
corresponding to (a) gold-silicon, (b) silicon-air, and (c) gold-air, with
$f=0.5$, and to (d) gold-silicon, (e) silicon-air, and (f) gold-air, with
$f=0.2$. The numerical values used in these graphs are $\lambda=1$ $\mu$m and
$R=180$ $\mu$m. Different curves correspond to different gap sizes of $H=100$
nm, $H=200$ nm, and $H=400$ nm.

The lateral Casimir-Lifshitz forces for the same layered structures as above
are shown in Figs. 3a-c for the symmetric case with $f=0.5$. In this case, we
have assumed $R=180$ $\mu$m and $\lambda=1$ $\mu$m. Similar to the previous
study, three different compositions of gold-silicon, silicon-air, and gold-air
are considered each at three different gap sizes of $H=100$ nm, $H=200$ nm,
and $H=400$ nm. The lateral force is found to oscillate as a function of the
lateral displacement, reminiscent of the lateral Casimir force that is induced
by geometrical corrugations GK ; lateral-exp . The shape of the oscillatory
function approaches a sinusoidal behavior as the gap size increases,
consistent with the fact that higher harmonics do not contribute to the force
in that limit as also seen in geometrical lateral Casimir effect Emig-exact .
The amplitude of the oscillations increases by decreasing the gap size as well
as the contrast between the dielectric properties of the two regions.
Numerically, we find an amplitude of 0.5 pN for gold-silicon, 8 pN for
silicon-air, and 12 pN for gold-air, at the closest separation of $H=100$ nm.

Figure 3d-f show the lateral Casimir-Lifshitz forces between the same types of
structures as above, for the asymmetric case of $f=0.2$. Similarly, the
profiles of the lateral force are noticeably asymmetric, with the asymmetry
weakening as the gap size is increased and the shape of the profile approaches
that of a sinusoidal function (single harmonic). We also see comparatively
more significant enhancement of the amplitude of the oscillatory behavior as a
function of the lateral displacement. The amplitude of the oscillations is
found as 0.3 pN for gold-silicon, 5 pN for silicon-air, and 7 pN for gold-air,
at the closest separation of $H=100$ nm.

##  V Discussion

In this paper, we have proposed a mechanism by which it is possible to create
a lateral Casimir-Lifshitz force as well as controlled modulations in the
normal Casimir-Lifshitz force without geometrical corrugations. A coupling
similar to what exists in the case of corrugated surfaces gives rise to these
oscillatory forces, namely identical modes of the dielectric patterns couple
across the gap to generate a macroscopic coherence in the fluctuations. The
generic features of these oscillatory forces are very similar to those of the
forces caused by corrugations; the effect is stronger and involves more
harmonics at closer separations, while it weakens and only involves the lowest
mode of the pattern in the dielectric contrast at larger separations.

While the difference in the dielectric properties of the materials controls
the general strength of the above results, comparison between Fig. 2 and 3
shows that the modulations in the normal force are more strongly affected by
the contrast in the dielectric properties. The choice of air/vacuum as one
component also allows us to make predictions about geometrical features with
large corrugation amplitudes, which provides an approximation scheme for the
non-perturbative geometrical regime.

In the present calculations we have only used the second order terms in the
dielectric contrast perturbative series. Higher order terms shown in Eq. (8)
will introduce coupling between different modes of the dielectric pattern in a
systematic way, as imposed by the overall conservation of the sum of all
wavevectors (momenta). While the present is aimed at showing in terms of
tractable calculations, one can in principle carry out the calculation of the
Casimir-Lifshitz interaction in such dielectric heterostructures using
numerical diagonalization methods valery .

Controlled interactions between dielectric heterostructures with smooth outer
surfaces could be very useful in practical applications because it will help
avoid the complications of bringing surfaces with geometrical protrusions
close to each other while avoiding contact between them and controlling their
separations. Moreover, it is much easier to pattern dielectric properties of
materials in a controlled way than it is to shape them with the high precision
that is needed for Casimir effect type experiments.

###### Acknowledgements.

The authors thank the ESF Research Network CASIMIR for providing excellent
opportunities for discussion on the Casimir effect and related topics. This
work was supported by EPSRC under Grant EP/E024076/1.

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