# Advocating Feedback Control for Human-Earth System Applications

Guido Cavraro This work was authored by the National Renewable Energy
Laboratory, operated by Alliance for Sustainable Energy, LLC, for the U.S.
Department of Energy (DOE) under Contract No. DE-AC36-08GO28308. Funding
provided by the NREL Laboratory Directed Research and Development Program. The
views expressed in the article do not necessarily represent the views of the
DOE or the U.S. Government. The U.S. Government retains and the publisher, by
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###### Abstract

This paper proposes a feedback control perspective for Human-Earth Systems
(HESs) which essentially are complex systems that capture the interactions
between humans and nature. Recent attention in HES research has been directed
towards devising strategies for climate change mitigation and adaptation,
aimed at achieving environmental and societal objectives. However, existing
approaches heavily rely on HES models, which inherently suffer from
inaccuracies due to the complexity of the system. Moreover, overly detailed
models often prove impractical for optimization tasks. We propose a framework
inheriting from feedback control strategies the robustness against model
errors, because inaccuracies are mitigated using measurements retrieved from
the field. The framework comprises two nested control loops. The outer loop
computes the optimal inputs to the HES, which are then implemented by
actuators controlled in the inner loop. Potential fields of applications are
also identified.

##  I Introduction

A significant effort has been dedicated to understand how climate changes
because of human actions and how to limit the global warming, e.g., through
decarbonization paths and policies to. This field is extremely challenging
because it lies in the intersection of climate science, economics,
engineering, and social science. Classically, research on human systems and
the Earth system was conducted separately, even though humans and the
environments naturally interact and form a single complex system referred to
as HES [1], see Figure 1.

Human systems encompass various aspects of society, e.g., economic and social
dynamics. Earth systems typically describe the interaction between the climate
and the global scale biogeochemistry. Even though Earth science has always
recognized that humans are an important component, key characteristics of the
Anthropocene, e.g., human agency and social and economic networks, have not
been dynamically represented in classic Earth systems. Capturing these
dynamics in a new generation of Earth system models allow to address a number
of critical questions about socio-ecological interactions [2]. HES science
adopt a holistic approach to capture interactions and _feedbacks_ within and
between human systems yielding advantages such as the ability to represent
ecosystem dynamics more realistically at longer timescales, and provide
insights that cannot be generated using only Earth models. Surprisingly, the
literature on HES lacks substantial contributions from control engineers, who
possess expertise in modeling, handling, and controlling complex dynamical
systems.

Figure 1: Schematic representation of the HES. The Human and Earth system
blocks feature internal interactions and are linked by feedback.

Scientists developed several _integrated assessment models_ (IAMs), i.e.,
quantitative models that describe the global interplay between climate,
economy, production sectors, and society. IAMs integrate theories and
empirical results from several disciplines to understand how human development
and societal choices affect each other and the natural world, including
climate change [3]. IAMs constitute a fundamental tool for the development and
validation of decarbonization paths. This line of research has been so
impactful that Prof. W. Nordhaus, who developed one of the first IAMs (DICE),
was awarded the Nobel Prize. As we will show next, IAMs act in an _open loop_
fashion and provide results affected by model inaccuracies. This rises
questions about straightforwardly applying IAMs’ results in practical
applications [4]. Section II reports a brief discussion about IAMs.

Since feedback control features well-established properties of errors and
uncertainty rejection, we argue that a _closed loop_ approach can be useful to
overcome the limitations of IAMs. This work then suggests a feedback control
perspective for studying HESs. Precisely, in Section III, we propose a control
framework consisting of two loops. Section IV describes the outer control loop
whose goal is to obtain the optimal control inputs steering the HES system
toward desirable states, i.e., states meeting properties like a bounded
average temperature increase or a minimum global welfare level. Section V
proposes an inner control loop whose goal is to ensure that the aforesaid
optimal setpoints are implemented in the HES .

Notation. Vectors are denoted by lower case boldface letters, respectively.
Calligraphic symbols denote sets; $\mathbb{R}$ denotes the set of real and
complex numbers.

##  II IAMs and the current approach

We can broadly group the IAMs into simple and complex IAMs. The first ones
rely on simplified equations and are often used to calculate the _social cost
of carbon_ , a measure of the quantifiable costs and benefits of emitting one
additional ton of $\text{CO}_{2}$. Among them, we mention DICE, PAGE, and
FUND. DICE is mainly used to compute optimal emissions pathways and the social
cost of carbon [5]. PAGE includes regional details and expresses all the main
inputs as probability distributions so that the risk of climate change could
more easily be appreciated [6]. FUND study the role of international capital
transfer in climate policies and features more regional and sectorial detail
[7].

In contrast, complex IAMs like GCAM [8], IMAGE [9], WITCH [10], MERGE [11],
and EPPA [12], are built on interconnected modules representing various
aspects of HESs, including the global economy, energy systems, land usage,
climate, and carbon cycles. Depending on the purpose of the IAM, the
corresponding climate and carbon modules vary in complexity and resolution,
e.g., see MAGICC [13] and HECTOR [14].

Carbon cycle simulators essentially relate carbon emissions from anthropic
activities and natural processes, carbon absorption from soil, vegetation, and
ocean, radiating forcing, and temperature. Of particular interest is how they
model _radiative forcing_ , i.e., the change in net radiation balance (solar
plus long-wave) of the surface-troposphere system. Carbon cycle simulators
typically express radiative forcing as a function of atmospheric
concentrations of greenhouse gases (GHGs), e.g., $\text{CO}_{2}$, Ozone,
Methane, Nitrus oxide. Even though climate models in different IAMS may
consider different spatial or time scales, in principle they model the global
carbon cycle and compute future concentrations of GHGs from given emissions,
calculate the global mean radiative forcing from GHG concentrations, and
convert the radiative forcing to global mean temperature [14].

Climate engineering, or geoengineering, is the intervention in the climate
system to counter climate change by rectifying the radiative forcing imbalance
[15, 16]. Climate engineering is divided into two broad categories: _Carbon
Dioxide Removal_ (CDR) and _Solar Radiation Management_ (SRM). CDR affects the
link from emissions to concentrations, SRM aims to break the link from
concentrations to temperatures.

CDR aims to remove $\text{CO}_{2}$ from the atmosphere by creating or
enhancing carbon sinks. CDR encompasses a variety of solutions, e.g., land
carbon sink enhancement via afforestation and reforestation, and ocean carbon
sink enhancement through increasing ocean alkalinity, and iron fertilization.
The effect of CDR options decays over time, because of the response of the
ocean and land carbon reservoirs to atmospheric perturbations, and if the
carbon storage is not permanent.

SRM aims to decrease the radiative forcing by reducing the amount of solar
radiation absorbed. SRM could be achieved through sunshades in space,
stratospheric aerosols (i.e., injection of sulphate aerosols into the lower
stratosphere emulating large volcanic eruptions), enhancing cloud albedo
(e.g., increasing the reflectivity of low level marine stratiform clouds), or
surface albedo (e.g., increasing grassland and cropland albedo). SRM methods
have common issues; in particular, reducing incoming solar radiation does not
ameliorate ocean acidification caused by rising atmospheric $\text{CO}_{2}$.

It is worth pointing out that climate engineering and its effect on the
climate system are typically not explicitly accounted for in the mentioned
IAMs. Indeed, CDR technologies are included only in a few attempts, e.g., see
(in GCAM) [17] or (in DICE) [18], while SRM is not yet, even though it would
clearly influence the radiative forcing. Experimental and synthetic data from
detailed climate simulators, see [19], can be used to learn to explicitly
model the effect of geoengineering on the climate system, e.g., the effect of
SRM technologies on radiative forcing.

##  III A feedback control framework

This paper suggest to adopt a feedback control approach in the HESs study. We
proposes a control framework that, using measurements from the HES, output
control setpoints performing, e.g., climate change adaptation and mitigation
actions. The framework is constituted by:

  1. 1.

An outer and slower control loop whose goal is to identify policies, i.e.,
functions that, given the state of the HES, prescribe the optimal action.

  2. 2.

An inner and faster control loop whose goal is to actuate the setpoints
provided by the aforesaid policies.

To formalize the proposed framework, following a control engineering
perspective, the states of the HES and the control inputs need to be
identified. Let the vectors $\mathbf{x},\mathbf{u}$ collect the states and the
inputs, respectively111For simplicity, we assume that the states are directly
measured, though a more general framework where states are not accessible can
be handled by adding a state estimator, e.g., a Kalman filter.. The state
$\mathbf{x}$, belonging to a state set $\mathcal{X}$, comprises the variables
needed to describe and predict the system behavior with reasonable accuracy.
States typically chosen in the literature include:

  1. 1.

_Climate variables_ , e.g., $\text{CO}_{2}$ concentrations in the atmosphere,
carbon stock in the soil and oceans, and the average temperature.

  2. 2.

_Socio-economic variables_ , e.g., the overall population, the world gross
product, the capital stock, the Renewable energy knowledge stock.

Control inputs $\mathbf{u}$, belonging to a state set $\mathcal{U}$, represent
the variables that can be directly regulated to steer the HES toward targeted
values and may include:

  1. 1.

_Climate inputs_ , e.g., $\text{CO}_{2}$ amount captured and stored by CDR
technologies, the amount of sprayed particles in the stratosphere, and
afforestation goals.

  2. 2.

_Socio-economic inputs_ , e.g., a carbon tax, the capital invested in the
production processes ([5]) or in renewable energy, the gross world product
growth rate [20].

Formally, the HES can be modeled by a vector function

$\mathbf{f}:\mathcal{X}\times\mathcal{U}\rightarrow\mathcal{X},(\mathbf{x},\mathbf{u})\mapsto\mathbf{f}(\mathbf{x},\mathbf{u}).$

Figure 2 graphically depicts the proposed control architecture. The controller
governing the outer control loop, denoted as $\mathbf{c}_{\text{out}}$,
computes and updates the HES control setpoint $\mathbf{u}^{*}$ based on the
HES state and on the implemented input $\mathbf{u}$. The inner loop
controller, denoted as $\mathbf{c}_{\text{in}}$, ensures that $\mathbf{u}^{*}$
is actually applied to the HES. The two control loops might act on two
different time scales:

  1. 1.

the outer loop on a slower one, since HESs evolve on several months/a few
years time intervals.

  2. 2.

the inner loop on a faster one, since, ideally, $\mathbf{u}^{*}$ should be
quickly implemented.

Notably, control theory gives the tools to: (i) handle the time scale
separation, e.g., by exploiting singular perturbation theory; (ii) provide
formal conditions for the stability of the closed loop system; (iii) provide
performance guarantees.

Figure 2: Block scheme representation of the proposed control framework.

##  IV Optimization of HES

The goal is to compute the action/intervention $\mathbf{u}^{*}$ for the HES
solving the multi-stage optimization problem

$\displaystyle\mathbf{u}^{*}:=\arg\min_{\mathbf{u}\in\mathcal{U}}\ $
$\displaystyle~{}\sum_{t=1}^{T}J(\mathbf{x}(t),\mathbf{u}(t))$ (1a)
$\displaystyle\mathrm{s.t.}\ $
$\displaystyle~{}\mathbf{x}(t+1)=\mathbf{f}(\mathbf{x}(t),\mathbf{u}(t))$ (1b)
$\displaystyle~{}\mathbf{g}(\mathbf{x}(t),\mathbf{u}(t))\leq 0$ (1c)

where the objective function has the form

$J:\mathcal{X}\times\mathcal{U}\rightarrow\mathbb{R},\quad(\mathbf{x},\mathbf{u})\mapsto
J(\mathbf{x},\mathbf{u}).$

The constraint (1b) captures the HES behavior. The inequality constraint (1c),
where

$\mathbf{g}:\mathcal{X}\times\mathcal{U}\rightarrow\mathbb{R}^{N},(\mathbf{x},\mathbf{u})\mapsto
g(\mathbf{x},\mathbf{u})$

maps the HES states and control inputs into $N$ constraints that describe
desirable system properties. For instance, $\mathbf{g}$ can capture effects
such as the global mean temperature increase or precipitation changes, effects
of climate change on renewable energy generations, access to food and water.

State-of-the-art IAMs obtain $\mathbf{u}^{*}$ by _numerically_ solving
problems like (1). Because of unavoidable errors and uncertainties in the
peculiar HES models they use, different IAMs arrive at strikingly different
solutions. Hence, the solution $\mathbf{u}^{*}$ is not necessarily the _true_
optimal setpoint and depends on the used HES model $\mathbf{f}$. Table I
reports the generation mixes computed by three IAMs for two _Shared
Socioeconomic Pathways_ (SSPs)222SSPs are scenarios of projected socioeconomic
global changes defined in the IPCC Sixth Assessment Report on climate change.
SSPs are used to derive GHG emissions under different climate policies. In
SSP1, consumption is oriented toward low material growth and lower resource
and energy intensity with a resulting low global average temperature rise.
SSP2 represents a scenario where social, economic, and technological trends do
not shift markedly from historical patterns. Even though environmental systems
degrade, overall the intensity of resource and energy use declines..
Structural differences between models lead to energy mixes that differ both in
the adopted technologies and in the amount of produced energy [4]. For these
reasons, there are even scientists claiming the very extreme statement that
IAMs are _useless_ and create a perception of knowledge and precision without
scientific legitimacy, see [21, 4].

| GCAM | IMAGE | WITCH
---|---|---|---
SSP1 | Coal | 173.5 | 116.1 | 83.5
Oil | 180.9 | 54.5 | 164.1
Gas | 162.3 | 216.9 | 140.6
Nuclear | 2.3 | 0 | 16.2
Biomass | 94.5 | 137.4 | 129.2
Renewables | 161.9 | 181.2 | 196.7
Total Energy | 775.4 | 706.1 | 730.1
SSP2 | Coal | 431.2 | 509.8 | 234.7
Oil | 349.8 | 180.5 | 360.1
Gas | 209.9 | 215.4 | 301.6
Nuclear | 61.9 | 11.9 | 42.2
Biomass | 109.1 | 116.5 | 136.8
Renewables | 87.2 | 123.4 | 202.9
Total Energy | 1249.1 | 1145.6 | 1278.3
TABLE I: Energy mixes proposed by three IAMs in [EJ], see [22].

From a system and control perspective, IAMs act in an _open loop_ fashion:
That is, the control input $\mathbf{u}^{*}$ is computed by solving problem (1)
relying only on the considered HES model and on his initial conditions, see
Figure 3$a)$. The IAMs substantially different results can then be explained
by the well-known fact that open loop strategies are very sensitive to model
inaccuracies and uncertainties.

Conversely, problem (1) can be solved in a _closed loop_ fashion, e.g., by
utilizing tools from Model Predictive Control (MPC) [23], or feedback-based
optimization [24], see Figure 3$b)$. The HES dynamically evolves toward
$\mathbf{u}^{*}$ following the input computed by the control function

$\mathbf{c}_{\text{out}}:\mathcal{\mathcal{X}}\times\mathcal{U}\rightarrow\mathcal{U},\quad(\mathbf{x},\mathbf{u})\mapsto\mathbf{c}_{\text{out}}(\mathbf{x},\mathbf{u}).$

which updates the control input from measurements of the actual state and
inputs. We envision this approach providing benefits from the fact that closed
loop strategies:

  1. 1.

have well-established properties of disturbance rejection: they actively
compensate model uncertainties, inaccuracies and disturbances using
measurements.

  2. 2.

update and adjust the control input using the information provided by the
measurements until they reach (asymptotically) the desired steady state value.

  3. 3.

can be tuned to obtain desired transitorial behavior, e.g., in terms of
setting time or overshoot.

For instance, adopting an MPC approach, problem (1) can be solved to obtain a
sequence $\mathbf{u}^{*}(1),\mathbf{u}^{*}(2),\dots,\mathbf{u}^{*}(T)$. Then,
only the first input $\mathbf{u}^{*}(1)$ is applied to the HES and the
optimization problem (1) is solved again based on the applied control setpoint
and the new measured state.

Figure 3: a) IAMs act in an open loop fashion. b) The closed loop approach.

Because the variables of interest, e.g., the average temperature, evolve
slowly, the outer loop controller acts and adjusts $\mathbf{u}^{*}$ on a time
scale of several months/a few years.

##  V Control of Actuators

_Actuators_ are agents interacting with the HES that are responsible for
implementing the optimal setpoints computed in the outer control loop.
Actuator is used here as an abstract word that in practice has different form
depending on the system where is acting. Examples includes

  1. 1.

in _climate systems_ carbon capture and storage facilities, plants for ocean
fertilization.

  2. 2.

in _socio-economic systems_ , policy and decision makers setting a carbon tax
or funding renewables.

We formally define the actuators map as the function

$\mathbf{a}:\mathcal{R}\rightarrow\mathcal{U},\quad\mathbf{r}\mapsto\mathbf{a}(\mathbf{r})$

that outputs the control inputs for the HES, where $\mathcal{R}$ is the set of
admissible reference signals. The goal is to create a sequence of inputs
$\\{\mathbf{u}(t)\\}$ converging to the desired value $\mathbf{u}^{*}$, see
Fig. 2. For example, consider an SRM application. The vector $\mathbf{u}^{*}$
will contain the desired albedo and the vector $\mathbf{r}$ will contain the
instructions for the actuators, e.g., the location and the amount of particles
sprayed by aircraft in the stratosphere. The work [19] proposes a
proportional-integral (PI) controller for an SRM application, where the goal
is to regulate the albedo by means of stratospheric aerosol injections. From
measurements of temperature, the PI scheme ensures the implementation of the
desired solar reduction. Other techniques could be applied to manage
actuators, like reinforcement learning [25].

Feedback-based optimization [24] looks a promising tool for controlling
actuators and for finding the best $\mathbf{r}$ implementing $\mathbf{u}^{*}$.
First, cast an optimization problem like

$\displaystyle\min_{\mathbf{r}\in\mathcal{R}}\ $
$\displaystyle~{}\phi(\mathbf{u}^{*},\mathbf{u},\mathbf{r})$ (2a)
$\displaystyle\mathrm{s.t.}\ $
$\displaystyle~{}\mathbf{u}=\mathbf{a}(\mathbf{r})$ (2b)

A meaningful objective function in (2a),
$\phi:\mathcal{U}\times\mathcal{U}\times\mathcal{R}\rightarrow\mathbb{R}$,
could penalize a weighted sum of the difference between $\mathbf{u}^{*}$ and
the actuator output and the cost of $\mathbf{r}$, e.g.,

$\phi(\mathbf{u}^{*},\mathbf{u},\mathbf{r})=\|\mathbf{u}^{*}-\mathbf{u}\|^{2}+\alpha\|\mathbf{r}\|^{2}$
(3)

with $\alpha$ being a positive constant. By plugging (2b) into (2a), problem
(2) becomes

$\min_{\mathbf{r}\in\mathcal{R}}\
\phi(\mathbf{u}^{*},\mathbf{a}(\mathbf{r}),\mathbf{r}).$

Instead of solving (2) numerically and apply directly its solution, feedback-
based optimization adopts an iterative approach, e.g., the projected gradient
descent algorithm

$\mathbf{r}(t+1)=\Big{[}\mathbf{r}(t)-\epsilon\frac{\partial\phi(\mathbf{u}^{*},\mathbf{u}(t),\mathbf{r}(t))}{\partial\mathbf{r}}\Big{]}_{\mathcal{R}}$
(4)

where $[\cdot]_{\mathcal{R}}$ represents the projection over $\mathcal{R}$ and
$\epsilon$ is a suitable constant that guarantees the convergence of (4). The
updated reference value $\mathbf{r}(t+1)$ is every time applied to the system.
The computation of $\partial\phi$ usually exploits measurements that represent
the feedback from the system, i.e., how the system reacted to the new
reference, and reduce the effect of modeling errors on $\mathbf{a}$. The inner
loop controller can then in general be expressed as the function

$\mathbf{c}_{\text{in}}:\mathcal{U}\times\mathcal{U}\times\mathcal{R}\rightarrow\mathcal{R},\quad(\mathbf{u}^{*},\mathbf{u},\mathbf{r})\mapsto\mathbf{c}_{\text{in}}(\mathbf{u}^{*},\mathbf{u},\mathbf{r}).$

Heed that other optimization algorithms have seen a feedback implementation,
e.g., primal-dual methods, dual ascent, ADMM. The iterative approach ensure
that the sequence $\mathbf{a}(\mathbf{r}(1)),\mathbf{a}(\mathbf{r}(2)),\dots$
converges (close) to $\mathbf{u}^{*}$.

##  VI Application Examples

This Section presents a few applications where the proposed control framework
could be applied. In each example, we identify the ingredients, e.g., the
state and control vectors $\mathbf{x}$ and $\mathbf{u}$, the HES model
$\mathbf{f}(\mathbf{x},\mathbf{u})$, the cost $J(\mathbf{x},\mathbf{u})$ that
can be plugged into (1) and (2) to formulate the problem of interest.

###  VI-A SRM via Stratospheric Aerosol Injection (SAI)

SAI seeks to reproduce the cooling effect of volcanic eruptions by injecting
sulfates into the stratosphere and forming a reflective aerosol. Denote by
$T(\psi)$ the _zonal-mean temperature_ at latitude $\psi$. SAI aims at shaping
$T(\psi)$ to meet climate objectives, e.g., keeping global warming below
$2^{\circ}$ C.

Recent studies propose to achieve different zonal average stratospheric
sulfate distributions for regulating the global mean temperature $T_{0}$, the
interhemispheric temperature gradient $T_{1}$, and the equator-to-pole
temperature gradient $T_{2}$. This is to decrease the average temperature and
preserve other important climate features like precipitation patterns.
$T_{0},T_{1},T_{2}$ can be computed as the projection of $T(\psi)$ into the
three Legendre polynomials [19]

$L_{0}(\psi)=1,\quad L_{1}(\psi)=\sin\psi,\quad
L_{2}(\psi)=1/2(3\sin^{2}\psi-1).$

The Legendre polynomials are also used as a basis to describe zonal average
stratospheric sulfate distributions which affect the radiative forcing [26].
Denote by $D(\psi)$ the Aerosol Optical Depths (AODs) at the latitude $\psi$.
The projections of $D(\psi)$ on the Legendre polynomials are denoted as
$D_{0},D_{1},D_{2}$, respectively. Acting on $D_{0},D_{1},D_{2}$ produces
controllable changes in $T_{0},T_{1},T_{2}$ [27]. Aerosols can be injected at
high altitude (roughly 5 km above the tropopause) at just four locations,
i.e., $30^{\circ}$ N, $15^{\circ}$ N, $15^{\circ}$ S, $30^{\circ}$ S in
latitude along $180^{\circ}$ longitude, to obtain an approximation of
stratospheric sulfate distributions expressed as any combination of the above
polynomials. For example, different combinations of injections in these four
locations can produce patterns of aerosol optical depths that are constant,
linear, and quadratic in the latitude [27, 28].

The goal is to affect the radiative forcing via SAIs to counter climate change
and support GHG emissions cut policies. The HES here is a coupled atmosphere-
ocean general circulation model capable of representing the formation and
interactions of stratospheric sulfate aerosol from source gases and the
radiative forcing, e.g., the fully-coupled Community Earth System Model (CESM)
[28]. The state $\mathbf{x}$ comprises the variables appearing in the HES
equations. Given that any zonal-mean temperature can be approximated as a
combination of the Legendre polynomials

$T(\psi)=T_{0}L_{0}(\psi)+T_{1}L_{1}(\psi)+T_{2}L_{2}(\psi).$

and after defining the control input
$\mathbf{u}=\begin{bmatrix}T_{0}&T_{1}&T_{2}\end{bmatrix}^{\top}$, an
interesting outer optimization problem aims to find the parameters for the
best target zonal-mean temperature $T^{*}(\psi)$. A meaningful metric could be
penalizing the control effort

$J(\mathbf{x}(t),\mathbf{u}(t))=\|\mathbf{u}(t)\|.$

to minimize human intervention. The function
$\mathbf{g}(\mathbf{x},\mathbf{u})$ models how $\mathbf{u}$ influences the
quantities of interest, e.g., temperatures, temperature gradients, and
precipitation patterns.

Actuators are aircraft injecting sulfates into the stratosphere. The goal of
the inner control loop is to have the optimal control input $\mathbf{u}^{*}$
implemented using SAI. The vector $\mathbf{r}$ collects then the amount of
particles to inject in the aforementioned four locations. Assume that the
function
$\boldsymbol{\varphi}:\mathcal{R}\rightarrow\mathcal{D},\mathbf{r}\mapsto\boldsymbol{\varphi}(\mathbf{r})$
maps injections into the AOD components $D_{0},D_{1},D_{2}$, with

$D(\psi)=D_{0}L_{0}(\psi)+D_{1}L_{1}(\psi)+D_{2}L_{2}(\psi).$

where $\mathcal{D}$ is the set of feasible $D_{0},D_{1},D_{2}$. Also, let
$\xi$ be the function modeling how AODs affect the temperature

$\boldsymbol{\xi}:\mathcal{D}\rightarrow\mathcal{U},\begin{bmatrix}D_{0}&D_{1}&D_{2}\end{bmatrix}^{\top}\mapsto\boldsymbol{\xi}(\begin{bmatrix}D_{0}&D_{1}&D_{2}\end{bmatrix}^{\top})$

The functions $\boldsymbol{\varphi}$ and $\boldsymbol{\xi}$ can be learned
from simulations [28]. The actuator map is given then by the composition

$\mathbf{a}(\mathbf{r})=\boldsymbol{\xi}(\boldsymbol{\varphi}(\mathbf{r}))$

The system feedback is captured by measurements of temperatures and AOD, which
can be performed using satellites. For instance, observations from the
Moderate Resolution Imaging Spectroradiometer (MODIS) on NASA’s Terra
satellite are already used to monitor aerosol amounts around the world.

A feedback-based optimization algorithm like (4) could then be developed to
solve problem (2) adopting the cost (3).

###  VI-B Developing sustainable economic pathways.

IAMs are extensively used to evaluate the climatic and social impacts of
economic growth scenarios. As mentioned earlier, different modeling choices
across IAMs lead to different solutions. The differences can be alleviated by
solving (1) using feedback control rather than numerically. For example,
consider the model used in DICE [5]. Society preferences are modeled by a
social welfare function that ranks different paths of consumption. The DICE
model assumes that economic and climate policies should be designed to
optimize the flow of consumption over time. The HES state $\mathbf{x}$
encompasses:

  1. 1.

_climate variables_ : Earth surface and oceans temperature, carbon stock in
the land and the oceans, $\text{CO}_{2}$ atmospheric concentration and
emissions;

  2. 2.

_socio-economic variables_ : the world population (denoted as $\pi(t)$) and
gross product, the productivity, the per-capita consumptions, the damage of
climate change.

The HES (1b) comprises twenty one nonlinear equations [5]; the control
$\mathbf{u}$ includes the emission control rate, which influences the cost of
emissions reductions, and capital consumption (denoted as $\gamma(t)$) and
investments. The outer optimization problem objective function models the
social welfare

$J(\mathbf{x}(t),\mathbf{u}(t))=\frac{\pi(t)\left(\frac{\gamma(t)}{\pi(t)}\right)^{1-\eta}}{1-\eta}\Psi(t).$

$J(\mathbf{x}(t),\mathbf{u}(t))$ represents the discounted sum of the per
capita consumption, where $\Psi(t)$ is the discount factor at time $t$ and
$\eta$ models the marginal utility of consumption. The welfare function is
increasing in the per capita consumption of each generation, with diminishing
marginal utility of consumption. The constraint (1c) captures desirable
outcomes like a threshold on the average temperature change.

Actuators here are policymakers. The outer control recommends the best control
actions $\mathbf{u}^{*}$ and updates $\mathbf{u}^{*}$ based on how the HES
reacts to the applied control setpoint. The actuator model $\mathbf{a}$ can be
obtained using tools from implementation theory [29].

###  VI-C Marine cloud brightening for coral reef preservation

In the former examples, the control $\mathbf{u}^{*}$ consists of actions that
should be implemented on a _global_ scale. This may raise concern about the
implementability of such control actions, as different Countries have
different priorities or values. Conversely, here we present a _local_
application. Scientists are testing if marine cloud brightening (MCB) could be
effective in cooling the Great Barrier Reef to preserve the coral population
[30]. The coral reefs and ecosystems are indeed endangered because of a
combination of increased ocean temperatures and acidification. MCB involves
spraying sea salt aerosols into marine stratocumulus to increase cloud
reflectivity. The proposed control framework can be used to plan and actuate
MCB interventions.

The HES model comprises climatic equations modeling the dynamics in the
atmosphere, ocean, and land in the region of interest. A recent work adopted
the fully-coupled Community Earth System Model (CESM) [31]. The state
$\mathbf{x}$ includes the climate variables appearing in the HES equations,
e.g., the atmospheric temperatures. The control input $\mathbf{u}$ represents
the surface albedo.

The outer loop finds $\mathbf{u}^{*}$ by solving an optimization problem whose
cost penalizes the deviation from a desired water temperature $\tau$. The
water temperature, denoted as $\rho$, depends on the HES state and the albedo
reduction. After modeling the water temperature as a function

$\rho:\mathcal{X}\times\mathcal{U}\rightarrow\mathbb{R},\quad(\mathbf{x},\mathbf{u})\mapsto\rho(\mathbf{x},\mathbf{u})$

a cost function that can be used in (1) is

$J(\mathbf{x}(t),\mathbf{u}(t))=\|\tau-\rho(\mathbf{x}(t),\mathbf{u}(t))\|.$

The optimal albedo reduction obtained by solving (1) will be physically
implemented. Actuators will be boats endowed with turbines that blow a mist of
seawater that ascends into the sky whose model is the function
$\mathbf{a}(\mathbf{r})$. The reference signal $\mathbf{r}$ regulates the
functioning of the turbines to obtain the desired albedo reduction. The system
feedback consists of aerosol measurements possibly taken by drones,
satellites, or other boats [30].

##  VII A numerical example

Here, we provide some numerical experiments to illustrate the benefit of
adopting a feedback approach when the model parameters are affected by
uncertainty in the context of evaluate and plan economic pathways for climate
change mitigation. For our tests, we considered the AYS model, which is a low-
complexity model incorporating climate change, welfare growth and energy
transformation [20].

The system’s state comprises three variables. $A$ [GtC] is the atmospheric
carbon stock in excess w.r.t. a pre-industrial level; $Y$ [$/yr] is the gross
world product (GWP) and is an indicator of the wealth of the society; $S$ [GJ]
is the renewable energy knowledge stock. It models the know-how for the
production of renewable energy and is identified with the past cumulative
production of renewables. Hence,
$\mathbf{x}^{\top}=\begin{bmatrix}A&Y&S\end{bmatrix}^{\top}$. The control
inputs are the growth index change $\beta$ [%$\ \text{yr}^{-1}$] which models
the GWP’s growth, and the break-even knowledge level $\sigma$ [GJ] at which
renewable and fossil extraction costs become equal which can be changed by
subsidizing renewables or introducing a carbon tax. That is,
$\mathbf{u}^{\top}=\begin{bmatrix}\beta&\sigma\end{bmatrix}^{\top}$. The
system obeys

$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},\mathbf{u}),\quad\mathbf{f}(\mathbf{x},\mathbf{u})=\begin{bmatrix}\frac{\sigma^{\rho}}{\sigma^{\rho}+S^{\rho}}\frac{Y}{\epsilon\phi}-\frac{A}{\tau_{A}}\\\
(\beta-\theta A)Y\\\
\frac{S^{\rho}}{\sigma^{\rho}+S^{\rho}}\frac{Y}{\epsilon}-\frac{S}{\tau_{S}}\end{bmatrix}$
(5)

Equation (5) represents an ODE parametrized in the climate change damage
coefficient $\theta\;[\text{yr}^{-1}\text{GtC}^{-1}]$, the energy efficiency
$\epsilon\;[\$\,\text{GJ}^{-1}]$, the fossil fuel combustion efficiency
$\phi\;[\text{GJ GtC}^{-1}]$, the time constants $\tau_{A}=50$ yr and
$\tau_{S}=50$ yr. The initial state and control inputs are

$A_{0}=840\text{ GtC},\,Y_{0}=7\cdot 10^{13}\$\text{ yr}^{-1},\,S_{0}=5\cdot
10^{11}\text{ GJ}$ $\beta_{0}=3\%\text{ yr}^{-1}\quad\sigma_{0}=5\cdot
10^{12}\text{ GJ},$

The system’s feedback consists of state measurements obtained once every two
years.

We assume that the goal is to track a desired trajectory for the level of
atmospheric $\text{CO}_{2}$. To that aim, we cast an optimization problem in
which the objective function is

$J(\mathbf{x},\mathbf{u})=\frac{A^{2}}{\lambda}+\frac{(\sigma-\sigma_{0})^{2}}{\mu}+\frac{(\beta-\beta_{0})^{2}}{\nu}+\mathbf{u}^{\top}\begin{bmatrix}w_{\beta}&0\\\
0&w_{\sigma}\end{bmatrix}\mathbf{u}$

with $\lambda,\mu,\nu,w_{\beta},w_{\sigma}$ positive parameters. A
discretization of (5) provides the equality (1b). The operational constraints
(1c) are replaced by the equations

$\displaystyle A,Y,S,\sigma\geq 0$ (6a) $\displaystyle A\leq 345,\quad Y\geq
4\cdot 10^{13}$ (6b)

Inequality (6a) captures hard constraints on states and control inputs.
Further, (6b) describes a target subspace ensuring a certain level of welfare
and in which the level of atmospheric $\text{CO}_{2}$ is reduced. To simulate
the fact that uncertainties affect the model, we assume that the available
model parameters are $\theta=8.57\cdot 10^{-5},\epsilon=147,\phi=4.7\cdot
10^{10}$ whereas their _true_ value can deviate up to 20%.

A numerical (open loop) solution to the problem provides the control setpoints
to track the desired trajectory of $A$. However, due to the parameter errors,
applying these control inputs results in higher levels of atmospheric
$\text{CO}_{2}$. Adopting instead a closed-loop approach (precisely,
implementing an MPC algorithm), we can track the desired trajectory of $A$ up
to an initial transitorial phase.

The results are reported in Figure 4. We can observe that the closed-loop
control input evolutions differ significantly from the open-loop one. Indeed,
the closed-loop solution prescribes control actions that are more drastic than
the open-loop one, which would yield a degraded trajectory of $A$ roughly
delayed by about five years w.r.t. the reference signal.

Figure 4: Left panel: the excess atmospheric carbon stock, Center panel: the
GWP’s growth rate. Right panel: the break-even knowledge level.

##  VIII CONCLUSIONS

We argue that control theory is an effective tool for HESs, e.g., to tackle
the unprecedented challenge of climate change. Even though HES science already
encompasses aspects from several disciplines, e.g., climate science,
economics, and social science, our goal is to add a (control) engineering
perspective in a field that has been historically mainly a prerogative of
economists or Earth and climate scientists. Feedback control allows us to find
the optimal HES inputs mitigating the effects of modeling uncertainties and
errors. This paper proposes a control framework that features an outer loop
and an inner control loop. The outer loop iteratively updates the control
inputs needed to steer the HES to desirable configurations; the inner control
loop ensures that the control setpoints computed in the outer loop are
implemented by actuators. Possible applications are also presented.

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# Square-densities, and volume forms

Anders Kock

## Introduction

The Greek geometers (Heron et al.) discovered a remarkable result, namely that
the area of a triangle in the plane can be expressed in terms of the lengths
of the three sides. Here, length and area are seen as non-negative numbers,
which involves, in modern terms, formation of absolute value and square root.
To express the notions and results involved without these non-smooth
constructions, one can express the Heron Theorem in terms of the squares of
the quantities in question: if $g(A,B)$ denotes the square of the length of
the line segment given by $A$ and $B$, the Heron formula says that the square
of area of triangle $ABC$ may be calculated by a simple algebraic formula out
the three numbers $g(A,B)$, $g(A,C)$, and $g(B,C)$. Explicitly, the formula
appears in (1) below. In modern terms, the formula is (except for a
combinatorial constant $-16^{-1}$) the determinant of a certain symmetric
$4\times 4$ matrix constructed out of three numbers; see (2) below. This
determinant, called the Cayley-Menger determinant, generalizes to simplices of
higher dimesions, so that e.g. the square of the volume of a tetrahedron
(3-simplex) $(ABCD)$ in space is given (except for a combinatorial constant)
by the determinant of a certain $5\times 5$ matrix constructed out of the six
square lengths of the edges of the tetrahedron.

The Heron method has the advantage that it is symmetric w.r.to permutations of
the $k+1$ vertices of a $k$-simplex. Also, Heron’s formula does not refer to
the vector- or affine structure of the space.

In case where the $n$-dimensional Euclidean space, in which the $k$-simplex
lives, is provided with an orthonormal coordinate system, one may identify the
space with $R^{n}$, with the standard inner product $\langle-,-\rangle$. Then
the square lengths, square areas, square volumes etc. of the simplices can
also be calculated in coordinates by an other well known and simple
expression: namely as $(1/k!)^{2}$ times the Gram determinant of a certain
$k\times k$ matrix which one may build from the coordinates of the vertices
$A_{0}$, $A_{1}$, …$A_{k}$ of the simplex. (The Gram determinant itself
expresses the square volume of the parallelepipedum spanned by the vectors
$A_{i}-A_{0}$.)

An important difference beween the two formulae is the $(k+1)!$-fold symmetry
in the Heron formula, where the Gram formula is only $k!$-fold symmetric,
because of the special role of $A_{0}$.

This Gram method of calculating the square-volumes has the advantage that it
it is easy to describe algebraically, and in particular, it is easy to
describe what happens if one changes the metric; this is needed, when dealing
with Riemannian manifolds, where the metric (the metric tensor) changes from
point to point.

We begin in Section 1 by recalling the classical case of Euclidean spaces,
i.e. finite dimensional vector spaces over $R$ equipped with a positive
definite inner product; equivalently, with a constant metric tensor. In
particular, we recall the comparison (standard, but non-trivial) between the
Heron and Gram calculations. In Section 2, we study Riemannian metrics on
manifolds, in synthetic, or combinatorial terms, and the resulting
“infinitesimal” simplices, and their square volume. In Section 3, we relate
the volume form of an $n$-dimensional Riemannian manifold to the volume of
certain infinitesimal $n$-simplices. This was also attempted in my [5]; I hope
that the present account will be less ad hoc.

### Conventions etc.

We use “SDG” as an abbreviation of “Synthetic Differential Geometry”, as
expounded in [3], say, and in particular in [4]; $R$ denotes the number line.
Elements in $R^{n}$ we think of as $n\times 1$ matrices (column vectors). The
word square-density  is used, where one in the positive definite case would
rather say “square length”, “square area”, “square volume” …. It is useful to
think in terms of the quantities occurring as being quantities whose physical
dimension is some power of length (measured in meter $m$, say), so that length
is measured in $m$, area in $m^{2}$, square area in $m^{4}$, etc. Tangent
vectors are not used in the following; they would have physical dimension of
$m\cdot t^{-1}$ (velocity).

##  1 Square volumes in Euclidean spaces

###  1.1 Heron’s formula

The basic idea for the construction of a square $k$-volume function goes, for
the case $k=2$, back to Heron of Alexandria (perhaps even to Archimedes); they
knew how to express the square of the area of a triangle $S$ in terms of an
expression involving only the lengths $a,b,c$ of the three sides:

$area^{2}(S)=t\cdot(t-a)\cdot(t-b)\cdot(t-c)$

where $t=\frac{1}{2}(a+b+c)$. Substituting for $t$, and multiplying out, one
discovers (cf. [2] §1.53) that all terms involving an odd number of any of the
variables $a,b,c$ cancel, and we are left with

$area^{2}(S)=-16^{-1}(a^{4}+b^{4}+c^{4}-2a^{2}b^{2}-2a^{2}c^{2}-2b^{2}c^{2}),$
(1)

an expression that only involves the squares $a^{2}$, $b^{2}$ and $c^{2}$ of
the lengths of the sides.

The expression in the parenthesis here may be written in terms of a
determinant of a $4\times 4$ matrix (described in (2) below, with $g(01)$
denoting $a^{2}$ etc.), which makes it easy to generalize from 2-simplices (=
triangles) to $k$ simplices, in terms of determinants of certain
$(k+2)\times(k+2)$ matrices, “Cayley-Menger matrices/determinants”; they again
only involve the square lengths of the $\binom{k+1}{2}$ edges of the simplex.

An oriented $k$-simplex in a space $M$ is a $(k+1)$-tuple of points (vertices)
$(x_{0},x_{1},\ldots,x_{k})$ in $M$. If $g:M\times M\to R$ satisfies
$g(x,x)=0$ and $g(x,y)=g(y,x)$ for all $x$ and $y$ (like a metric
$\operatorname{dist}(x,y)$, or its square), one may construct a
$(k+2)\times(k+2)$ matrix $C(x_{0},x_{1},\ldots,x_{k})$ by the following
recipe: take the $(k+1)\times(k+1)$ matrix whose $ij$th entry is
$g(x_{i},x_{j})$. It is symmetric, and has $0$s down the diagonal, by the two
assumption about $g$. Enlarge this matrix it to a $(k+2)\times(k+2)$\- matrix
by bordering it with $(0,1,\ldots,1)$ on the top and and on the left. The case
$k=2$ is depicted here (writing $g(ij)$ for $g(x_{i},x_{j})$ for brevity; note
$g(01)=g(10)$ etc., so that the matrix is symmetric.

$\left[\begin{array}[]{cccc c}0&1&1&1\\\ 1&0&g(01)&g(02)\\\ 1&g(10)&0&g(12)\\\
1&g(20)&g(21)&0\end{array}\right]$ (2)

(The indices of the rows and columns are most conveniently taken to be
$-1,0,1,2$.)

This is the Cayley-Menger matrix for the simplex, and its determinant is the
Cayley-Menger determinant. Heron’s formula then says that the value of this
determinant is, modulo the “combinatorial” factor $-16^{-1}$, the square of
the area of a triangle with vertices $x_{0},x_{1},x_{2}$, as expressed in
terms of square $g(x_{i},x_{j})$ of the distances between them. Similarly for
(square-) volumes of higher dimensional simplices. Note that no coordinates
are used in the construction of this determinant.

The general formula is that the square of the volume of a $k$-simplex is
$-(-2)^{k}\cdot(k!)^{-2}$ times the determinant, e.g. for $k=1,2$, and $3$,
the factors are $2^{-1}$, $-16^{-1}$, and $288^{-1}$, respectively.

###### Proposition 1.1

The Cayley-Menger determinant for a $k$-simplex is invariant under the
$(k+1)!$ symmetries of the vertices of the simplex.

Proof. Interchanging the vertices $x_{i}$ and $x_{j}$ has the effect of first
interchanging the $i$th and $j$th column, and then interchanging the $i$t and
$j$th row of the matrix. Each of these changes will change the determinant by
a factor $-1$.

We shall in the following denote the square volume of a $k$-simplex $X$ in
$R^{n}$, as calculated by the Heron-Cayley-Menger formula, by $vol^{2}(X)$. We
shall later generalize it by replacing the standard inner product on $R^{n}$
by an arbitrary symmetric bilinear form $G$ on $R^{n}$. Such $G$ is
represented by a symmetric $n\times n$ matrix, likewise denoted $G$, and the
generalized version is denoted $vol^{2}_{G}(X)$ or $vol^{2}_{g}(X)$, where $g$
is the square length derived from $G$. The standard inner product on $R^{n}$
corresponds to the identity matrix $I$; and $vol^{2}(X)$ is therefore also
denoted $vol^{2}_{I}(X)$.

###  1.2 Gram’s formula

Given a $k$-simplex $X=(x_{0},x_{1},\ldots,x_{k})$ in $R^{n}$. Then we may
form the matrix $Y$, which is an $n\times k$ matrix with columns $x_{i}-x_{0}$
($i=1,\ldots,k$) where we think of the $x_{i}$s ($i=0,1,\ldots,k$) as $n\times
1$ matrices (column matrices). The Gram matrix of the simplex $X$ is then the
$k\times k$ matrix $Y^{T}\cdot Y$. Its determinant is the Gram determinant of
the simplex.

This determinant likewise has a volume theoretic significance: it gives the
square of the volume of the parallelepipedum spanned by the $k$ vectors
$x_{i}-x_{0}$ ($i=1,\ldots,k$). So geometrically, it depends on the geometric
construction of the parallelograms out of which the parallelepipedum is
constructed.

The following Proposition is only included for a comparison with the issue of
$(k+1)!$ symmetry of the formulae.

###### Proposition 1.2

The Gram determinant for a $k$-simplex is invariant under the $(k+1)!$
symmetries of the $k$-simplex.

Proof. Interchanging $x_{i}$ and $x_{j}$, for $i$ and $j\geq 1$ impies an
interchange of the corresponding columns in the $Y$-matrix, and this
interchanged matrix comes about by multiplying $Y$ on the right by the
$k\times k$ matrix $S$ obtained from the unit matrix by interchanging its
$i$th and $j$th column. This $S$ has determinant $-1$. So $S^{T}\cdot
Y^{T}\cdot Y\cdot S$ has same determinant as $Y^{T}\cdot Y$. Interchanging
$x_{0}$ and $x_{1}$ in the simplex corresponds to multiplying the $Y$-matrix
on the right by the matrix $S_{j}$, obtained from the unit $k\times k$ matrix
by replacing its first row by the row $(-1,-1,\ldots,-1)$. The matrix $S$ has
determinant $-1$, so $S^{T}\cdot Y^{T}\cdot Y\cdot S$ has same determinant as
$Y^{T}\cdot Y$. Similarly for exchanging $x_{0}$ by $x_{j}$.

###  1.3 Comparison formula, and $vol^{2}(X)$

In the vector space $R^{n}$, with its standard inner product, it makes sense
to compare the values of the formulas for square volume of a $k$-simplex
$X=(x_{0},x_{1},\ldots,x_{k})$. Let $C$ denote the $(k+2)\times(k+2)$ matrix
(Cayley-Menger) formed by the square distances between the vertices, as
described above, and let $Y^{T}\cdot Y$ be the Gram $k\times k$ matrix,
likewise described above. There is a known relation between their determinants

$\frac{1}{-(-2)^{k}}\det(C)=\det(Y^{T}\cdot Y)^{2}.$ (3)

For a proof, see reference [10]. Both sides express $(k!)^{2}$ times the
square $vol^{2}(X)$ of the volume of the simplex $X$ (or express the square of
the volume of the parallelepipedum spanned by the column vectors $x_{i}-x_{0}$
in the matrix $Y$). But note that the right hand side only makes sense in a
coordinatized situation, so in the following, $vol^{2}(X)$ is used for the
more general Heron formula for the square volume of a simplex, as calculated
in terms of the Cayley-Menger determinant $C$. In $R^{n}$ the relationship is
thus, for a $k$-simplex $X$ in $R^{n}$,

$vol^{2}(X)=\frac{1}{k!^{2}}\det(Y^{T}\cdot Y)$ (4)

###  1.4 Change of metric

The formula (3) refers to the vector space $R^{n}$ with the standard inner
product. The Gram formula is easy to extend when one wants to replace this
inner product on $R^{n}$ by another, and square length is given in terms of
this new inner product $\langle-,-\rangle_{G}$. We identify the new inner
product by a symmetric $n\times n$ matrix $G$, with $\langle u,v\rangle_{G}$
defined as the matrix product $u^{T}\cdot G\cdot v$. (The standard inner
product correpsonds to the identity $n\times n$ matrix $I$, so $\langle
u,v\rangle=u^{T}\cdot I\cdot v$.)

Given a $k$-simplex $X=(x_{0},x_{1},\ldots,x_{k})$ in $R^{n}$. Let $Y$ be the
$n\times k$ matrix whose $k$ columns are $x_{j}-x_{0}$, with $j=1,\ldots,k$.
Then the $k\times k$ matrix $Y^{T}\cdot G\cdot Y$ is the $G$-generalization
for the Gram matrix of the simplex, and its determinant is the square of the
volume, as defined by the metric $G$, of the parallelepipedum spanned by the
column vectors in $Y$.

Can we calculate the square volume of this simplex using the Heron-Cayley-
Menger method, when one uses the metric $G$ rather than the standard metric?
This question can be answered by yes, provided that $G$ is positive definite.
More precisely, we shall assume that $G$ can be written as a matrix product
$H^{T}\cdot H$, with $H$ another $n\times n$ matrix. (Over the real numbers
${\mathbb{R}}$, this property characterizes positive definite quadratic forms,
cf. e.g. [7] XI Prop. 6.)

Let us write $vol^{2}_{G}(X)$ or $vol^{2}_{G}(x_{0},\ldots,x_{k})$ for the
square volume of the $k$-simplex $X$ spanned by the $x_{i}$s, when the metric
$G$ is used, so $vol^{2}_{I}(x_{0},\ldots,x_{k})$ is the square volume of the
simplex for the standard metric.

###### Proposition 1.3

If an $n\times n$ matrix $G$ can be written $G=H^{T}\cdot H$ for an $n\times
n$ matrix $H$, then ($G$ is symmetric and) we have for $vol^{2}_{G}$

$vol^{2}_{G}(x_{0},\ldots,x_{k})=vol^{2}_{I}(H\cdot x_{0},\ldots,H\cdot
x_{k}),$

and as such may be calculated in terms of the standard Cayley-Menger
determinant.

Proof. We have for any $y\in R^{n}$ that $H(y)^{T}=(H\cdot y)^{T}=y^{T}\cdot
H^{T}$, so

$y^{T}\cdot G\cdot y=y^{T}\cdot H^{T}\cdot H\cdot y=(H\cdot y)^{T}\cdot(H\cdot
y),$

so letting $y$ range over $x_{i}-x_{j}$ ($i,j=0,1,\ldots,k$), the left hand
side calculates the left hand expression; the right hand expression may be
calculated in terms of the Cayley-Menger determinant $C$).

We have, for $G$ and $H$ as in the Proposition, that

$\det(G)=\det(H)^{2}$ (5)

so that if $\det(G)$ is invertible, there are exactly two numbers which
deserve the name $\sqrt{\det(G)}$.

##  2 Infinitesimal simplices

As in [4], say, we consider the following kind of structure on an object $M$
in a category ${\mathcal{E}}$ with finite inverse limits $M$, namely
subobjects of $M\times M$,

$M_{(0)}\subseteq M_{(1)}\subseteq M_{(2)}\ldots\subseteq M\times M,$

each of the $M_{(r)}$s being a reflexive and symmetric relation, with
$M_{(0)}$ being the equality relation. We have in mind the “$r$th
neighbourhood of the diagonal” of $M$, as considered in algebraic geometry, or
the “prolongation spaces” of manifolds as considered in e.g. [6]. Except for
$M_{0}$, these relations are not transitive.

We use the well known “synthetic” language to express constructions in
categories ${\mathcal{E}}$ with finite limits, in “elementwise” terms111Recall
that a generalized element of an object $M$ in a category ${\mathcal{E}}$ is
just an arbitrary map in ${\mathcal{E}}$ with codomain $M$; see [3] II.1, [8]
V.5, or [9] 1.4., in particular we consider for a natural number $k$ the
object of $r$-infinitesimal $k$-simplices in $M$, meaning the subobject of
$M\times M\times\ldots\times M$ ($k+1$ times) consisting of $k+1$-tuples
$(x_{0},x_{1},\ldots,x_{k})$ of elements of $M$ with $(x_{i},x_{j})\in
M_{(r)}$ for all $i,j=0,1,\ldots,k$; such a $k+1$-tuple, we shall call an
$r$-infinitesimal $k$-simplex.

Note that the question of whether a $k$-simplex is $r$-infinitesimal only
depends on the “edges” $(x_{i},x_{j})$ (face-$1$-simplices) of the simplex,
equivalently, it depends on the 1-skeleton of the simplex. We shall consider
also a further condition on $k$-simplices which depends on the 2-skeleton,
i.e. it is a condition on the face-$2$-simplices of the simplex. We shall use
the predicate “thin”; this kind of condition was introduced by F. Bár in his
thesis [1]; see Section 4. This condition will be important in Section 3
below. We shall only use the predicate “thin” for the case $r=2$, so a thin
$k$-simplex is in particular 2-infinitesimal.

We shall, as in [4], write $x_{i}\sim_{r}x_{j}$ for $(x_{i},x_{j})\in
M_{(r)}$.

For $r=1$ and $r=2$, we shall consider maps from the object of
$r$-infinitesimal $k$-simplices to $R$ which have the property that they
vanish if $x_{i}=x_{j}$ for some $i\neq j$. For $r=1$, combinatorial
differential $k$ forms $\omega$ have this property: they vanish if
$x_{i}=x_{j}$. (In the context of SDG, such maps are automatically alternating
with respect to the $(k+1)!$ permutations of the $x_{i}$s, see [4] Theorem
3.1.5.)

For $r=2$, such maps have not been considered much222For $r=2$ and $k=1$, such
things were in [4] 8.1 called “quadratic differential forms”., except for the
case where $k=1$, where Riemannian metrics $g$, in the combinatorial sense,
are examples of such maps; for this case, we think of $g(x_{0},x_{1})$ as the
square of the distance between $x_{0}$ and $x_{1}$. The $g$s of interest are
symmetric, $g(x_{0},x_{1})=g(x_{1},x_{0})$. In the context of SDG, we have

###### Proposition 2.1

Given $g:M_{(2)}\to R$ with $g(x,x)=0$ for all $x$. Then $g$ is symmetric iff
it vanishes on $M_{(1)}\subseteq M_{(2)}$.

Proof. It suffices to consider an $R^{n}$ chart around $x$; we consider the
degree $\leq 2$ part of the Taylor expansion of $g$ around $x$. Then $g$ is
given as $g(x,y)=C(x)+\Omega(x;y-x)+(y-x)^{T}\cdot G(x)\cdot(y-x)$, where
$C(x)$ is a constant, $\Omega$ is linear in the argument after the semicolon,
and $G(x)$ is a symmetric $n\times n$ matrix . To say that $g$ vanishes on the
diagonal $M_{(0)}$ (i.e. $g(x,x)=0$ for all $x$) is equivalent to saying that
$C(x)=0$ for all $x$. We now compare $g(x,y)$ and $g(y,x)$; we claim

$(y-x)^{T}\cdot G(x)\cdot(y-x)=(y-x)^{T}\cdot G(y)\cdot(y-x).$ (6)

Taylor expanding from $x$ the $G(y)$ on the right hand side gives that this
the difference between the two sides is $(y-x)\cdot dG(x;y-x)\cdot(y-x)$ which
is trilinear in $y-x$, and therefore vanishes, since $x\sim_{2}y$. So we have
that if $\Omega$ vanishes, then $g$ is symmetric; vice versa, if $g$ is
symmetric, its restriction to $M_{(1)}$ is likewise symmetric, and (being a
differential 1-form), it is alternating, so the $\Omega$-part vanishes, which
in coordinate free terms says: $g(x,y)=0$ for $x\sim_{1}y$.

For the number line $R$, $(x_{0},x_{1})\in R_{(2)}$ iff $(x_{1}-x_{2})^{3}=0$,
and the map $g$ given by $g(x_{0},x_{1}):=(x_{1}-x_{0})^{2}$ is a map as
described in the Proposition. In fact, it is the restriction of the standard
“square-distance” function $R\times R\to R$.

So we recall, respectively pose, the following definitions, corresponding to
$r=1$ and $r=2$. Let $M$ be a manifold.

###### Definition 2.2

A (combinatorial) differential $k$-form on $M$ is function $\omega$ from the
set of 1-infinitesimal $k$-simplices in $M$ to $R$, which is alternating with
respect to the $(k+1)!$ permutations of the $x_{i}$s, and which vanishes if
$x_{i}=x_{j}$ for some $i\neq j$.

###### Definition 2.3

A $k$-square-density on $M$ is an $R$-valued function $\Sigma$ from the set of
$2$-infinitesimal $k$-simplices in $M$ which is symmetric with respect to the
$(k+1)!$ permutations of the $x_{i}$s, and which vanishes if $x_{i}=x_{j}$ for
some $i\neq j$.

Note that for $k=1$, Proposition 2.1 gives that 1-square densities $g$ have
the property that they vanish not just on $M_{(0)}$ (the diagonal), but also
on $M_{(1)}$: $g(x,y)=0$ if $x\sim_{1}y$. So the notion of 1-square density
agrees with “(combinatorial) quadratic form”, as considered in [4], Section
8.1.

###  2.1 $k$-square-densities $\Sigma_{g}$ from $1$-square-densities $g$

We shall argue that the Cayley-Menger determinants define square-densities. We
already argued above that these determinants are symmetric: the value does not
change when interchanging $x_{i}$ and $x_{j}$. We have to argue for the
vanishing condition required. If $x_{i}=x_{j}$, then
$g(x_{i},x_{m})=g(x_{j},x_{m})$ for all $m$, and this implies that the $i$th
and $j$th rows in the matrix are equal, which implies that the determinant is
$0$.

We denote the $k$-square-density corresponding to a $1$-square-density by
$\Sigma_{g}$ (when $k$ is understood from the context).

###  2.2 $k$-square-densities $\Sigma^{\omega}$ from differential $k$-forms
$\omega$

Essentially this is the process of squaring (in $R$) the values, so it is
tempting to denote the square-density $\Sigma^{\omega}$ which we are aiming
for, by $\omega^{2}$. However, the values of a differential form are always
numbers of square 0. We do get, however, a well defined $k$-square-density out
of a differential $k$-form by a two step procedure: 1) to extend the given $k$
form $\omega$ to a suitable function $\overline{\omega}$, to allow as inputs
2-infinitesimal $k$-simplices; and then 2) squaring $\overline{\omega}$
valuewise. “Suitable” implies in particular that the squared function
$\overline{\omega}$ is a $(k+1)!$-symmetric function when applied to
2-infinitesimal $k$-simplices. We shall prove that such an extension is
possible; but is not unique. It depends on choosing a coordinate chart.

The question is local, so let us assume that the manifold $M$ is an open
subset of $R^{n}$. Then the $k$-form $\omega$ is given by a function
$\Omega:M\times(R^{n})^{k}\to R$, such that

$\omega(x_{0},x_{1},\ldots,x_{k})=\Omega(x_{0};x_{1}-x_{0},x_{2}-x_{0},\ldots,x_{k}-x_{0})$

where for each $x_{0}\in M$, the function
$\Omega(x_{0};-,\ldots,-):(R^{n})^{k}\to R$ is $k$-linear and alternating in
the $k$ arguments; these arguments are arbitrary vectors in $R^{n}$, in
particular, they may be of the form $x_{i}-x_{0}$ for $x_{i}\sim_{2}x_{0}$, so
the restriction of $\Omega(x_{0};x_{1}-x_{0},x_{2}-x_{0},\ldots,x_{k}-x_{0})$
to the the set of 2-infinitesimal 2-simplexes defines an extension
$\overline{\omega}$ of $\omega$, so

$\overline{\omega}(x_{0},x_{1},\ldots,x_{k}):=\Omega(x_{0};x_{1}-x_{0},x_{2}-x_{0},\ldots,x_{k}-x_{0})$
(7)

In this form, the fact that $\omega$ is alternating w.r.to the $k!$
permutations of the $x_{i}$s ($i=1,\ldots,k$) can be read of from the fact
that $\Omega(x_{0};\ldots)$ is alternating. It is also alternating w.r.to
permutations involving $x_{0}$, as long as the $x_{i}$s are $\sim_{1}x_{0}$;
this can be seen from seen from a Taylor expansion argument, see the proof of
Theorem 3.1.5 in [SGM] (or: restrict the argument below to the case of
1-infinitesimal simplices). Now if we use $\Omega$ to construct the extension
$\overline{\omega}$, to 2-infinitesimal $k$-simplices, the constructed
$\overline{\omega}$ will still be alternating w.r.to permutations of the
$x_{i}$s for $i\geq 0$, but the Taylor expansion argument mentioned fails for
the interchange of, say, $x_{0}$ and $x_{1}$: we cannot conclude that
$\Omega(x_{1};x_{0}-x_{1},\ldots)=-\Omega(x_{0};x_{1}-x_{0},\ldots)$. This
failure get repaired by valuewise squaring:

###### Proposition 2.4

For any 2-infinitesimal $k$-simplex $(x_{0},x_{1},\ldots,x_{k})$, we have

$\Omega(x_{1};x_{0}-x_{1},x_{2}-x_{1},\ldots)^{2}=\Omega(x_{0};x_{1}-x_{0},x_{2}-x_{0},\ldots)^{2}.$

Proof. We shall only do the case $k=1$. (For the more general case, the
further argument is essentially the same as in the proof of Proposition 1.2
above.) First, we have by a Taylor expansion from $x_{0}$

$\Omega(x_{1};x_{0}-x_{1})=\Omega(x_{0};x_{0}-x_{1})+d\Omega(x_{0};x_{1}-x_{0};x_{0}-x_{1})$
$+\mbox{ a $d^{2}\Omega(x_{0};\ldots)$ term, trilinear in }x_{1}-x_{0}.$

The trilinear term vanishes, because $x_{1}\sim_{2}x_{0}$. Now we square, and
get

$\Omega(x_{1};x_{0}-x_{1})^{2}=\Omega(x_{0};x_{0}-x_{1})^{2}+2\cdot\Omega(x_{0};x_{0}-x_{1})\cdot
d\Omega(x_{0};x_{1}-x_{0},x_{1}-x_{0})$ $+\mbox{ a term
$(d\Omega(x_{0};\ldots))^{2}$, quadrilinear in }x_{1}-x_{0}.$

The quadrilinear term vanishes because $x_{1}\sim_{2}x_{0}$, but also the term
$\Omega\cdot d\Omega$ vanishes, because it is trilinear in $x_{1}-x_{0}$. So
we get

$\Omega(x_{1};x_{0}-x_{1})^{2}=\Omega(x_{0};x_{0}-x_{1})^{2}=(-\Omega(x_{0};x_{1}-x_{0}))^{2}=\Omega(x_{0};x_{1}-x_{0})^{2},$

as desired.

We conclude that a differential $k$-form $\omega$ can be extended to an
$\overline{\omega}$ (whose input are 2-infinitesimal $k$-simplices), such that
$\overline{\omega}^{2}$ is $(k+1)!$\- symmetric. (Also, the extension
constructed also clearly has the property that it vanishes if $x_{i}=x_{j}$
for some $i\neq j$.)

###### Proposition 2.5

If two extended $k$-forms extend the same differential $k$-form $\omega$, the
two resulting square-densities agree.

This is proved by passing to a coordinate chart, in which case the proof comes
about by a simple counting of degrees of the polynomial functions expressing
the two forms and their difference. This in turn is best formulated in terms
of $k$-fold tensor products of symmetric algebras; it is given in full in the
Appendix.

Because of the Proposition, there is a well-defined “squaring” process,
leading from differential $k$-forms to $k$-square-densities on a manifold $M$:
extend the for form $\omega$, and square the result.

##  3 Riemannian structure in coordinates

The question of when a 1-square-density $g$ qualifies as a Riemannian metric
may be described in terms of a coordinate chart.333A classical description,
which is coordinate free, is possible in terms of the tangent bundle $T(M)$,
namely in terms of bilinear maps $T_{x}\times T_{x}\to R$. See [4] 8.1.3 for a
comparison. So we consider a manifold $M$ which is embedded as an open subset
of $R^{n}$ (elements of $R^{n}$ we write as $n\times 1$ matrices). A 1-square-
density $g$ on $M$ can in this case be given by a family by a family of
symmetric $n\times n$ matrices $G(x)$ (for $x\in M$), such that for
$x\sim_{2}y$,

$g(x,y)=(y-x)^{T}\cdot G(x)\cdot(y-x)$

(which equals $(y-x)^{T}\cdot G(y)\cdot(y-x)$ by (6)). To say that such $g$ is
positive definite is then to say that each $G(x)$ can be written $H^{T}\cdot
H$ for some $n\times n$ matrix $H=H(x)$, as in Proposition 1.3, and assumed to
be invertible.

Recall the relationship between $g$ and $G$: $g(y,z)=(z-y)^{T}\cdot
G(y)\cdot(y-z)$. We shall prove that for thin simplices, the $G(y)$ here may
be replaced by suitable $G(x)$:

###### Lemma 3.1

Let $(x,y,z)$ be a thin 2-simplex. Then $g(y,z)$ may be calculated by
$(z-y)^{T}\cdot G(x)\cdot(z-y)$.

Proof. Taylor expand $G$ from $x$; we have

$\begin{array}[]{lr}g(y,z)=(y-z)^{T}\cdot G(y)\cdot(y-z)&\\\ =[(y-z)^{T}\cdot
G(x)\cdot(y-z)]+[(y-z)^{T}\cdot dG(x;y-x)\cdot(y-z)]+&\\\ +\mbox{ higher terms
involving }d^{2}G(x;\ldots)\cdot(y-z)\end{array}$

but the term $(y-z)^{T}\cdot dG(x;y-x)\cdot(y-z)$ (as well as the “higher”
terms), vanish by the assumed thin-ness because they contain
$(y-x),(y-z),(y-z)$ in a trilinear way.

A Riemannian metric on a manifold $M$ allows us to describe the square volume
of any 2-infinitesimal $k$-simplex in $M$: use the (generalized) Heron
formula. If the $k$-simplex is $(x_{0},\ldots,x_{k})$, the Heron (Cayley-
Menger-) formula is an expression in the $g(x_{i},x_{j})$s. In a coordinatized
situation, it can therefore be calculated in terms of the
$(x_{j}-x_{i})^{T}\cdot G(x_{i})\cdot(x_{j}-x_{i})$. For $k\geq 2$, this
expression is in general rather complicated, because there are several
$G(x_{i})$s involved. However, if the simplex is thin, any one of these will
do for all; for instance, they may all be replaced by $G(x_{0})$: this is an
immediate consequence of Lemma 3.1.

We consider a Riemannian manifold $M,g$, with $M$ an open subset of $R^{n}$;
the Riemannian metric $g$ on $M$ is not assumed to be the one induced by the
standard metric on $R^{n}$. It is given by positive definite $n\times n$
matrices $G(x)$, for $x\in M$.

###### Proposition 3.2

Let $X=(x_{0},\ldots,x_{k})$ be a thin $k$-simplex in $M\subseteq R^{n}$. Then

$vol^{2}_{g}(x_{0},\ldots,x_{k})=vol^{2}_{I}(H\cdot x_{0},\ldots,H\cdot
x_{k})$

where $H$ satisfies $H^{T}\cdot H=G(x_{0})$.

Proof. The left hand side is an expression in the square distances
$g(x_{i},x_{j})$, and they are given by expressions $(x_{i}-x_{j})^{T}\cdot
G(x_{i})\cdot(x_{i}-x_{j})$. For thin 2-infinitesimal simplices, we may by 3.1
replace all the $G(x_{i})$ occurring here by $G(x_{0})$. Thus, for a thin
$k$-simplex, we are back at the situation with a constant $G$, and can
therefore use Proposition 1.3.

Assume that $g$ is positive definite (in the above sense) at every $x\in M$,
so $G(x)=H(x)^{T}\cdot H(x)$ for some $n\times n$ matrix. Also
$\det(G(x))=(\det(H(x))^{2}$ so $\det(G(x))$ (as $x$ varies over $M$) defines
a function $W(g):M\to R$ that pointwise admits square roots. The assumed
square roots of them are only “half unique”; they are unique modulo sign. If
$M$ is connected and oriented, it is possible to choose a (smooth) function
whose square is $W$. This is possibly in exactly two ways, by assumption.
Choose one of these functions and denote it $\sqrt{\det(G)}$.

###### Theorem 3.3

Assume that $g$ is a Riemannian metric on an $n$-dimensional manifold $M$.
Then there exists locally on $M$ a differential $n$-form $\omega$ such that
$\Sigma_{g}$ and $\Sigma^{\omega}$ agree on all thin $n$-simplices; $\omega$
is (locally) unique up to sign, and deserves the name volume form for $g$.

Proof. Since the data and assertions in the statement do not depend on the
choice of a coordinate system, it suffices to prove the assertion in a
coordinate chart. So assume that $M\subseteq R^{n}$ is an open subset. We use
this coordinate chart to construct the extension $\overline{\omega}$ as
described in Subsection 2.2. Then $g$ is given in terms of the symmetric
positive definite $n\times n$ matrices $G(x)$ (for $x\in M$). Consider the
differential $n$-form $\omega$, as well as the extended $n$-form
$\overline{\omega}$, given by the formula

$\overline{\omega}(x_{0},x_{1},\ldots,x_{n}):=\frac{\sqrt{\det
G(x_{0})}}{n!}\cdot\det(x_{1}-x_{0},\ldots,x_{n}-x_{0})$

for any 2-infinitesimal $n$-simplex $X$. Let $Y$ denote the $n\times n$ matrix
with $x_{j}-x_{0}$ as its $j$th column. Then squaring the defining equality
for $\overline{\omega}$ gives

$\overline{\omega}^{2}(X)=\frac{\det G(x_{0})}{n!^{2}}\cdot(\det
Y)^{2}=\frac{1}{n!^{2}}\det(Y^{T}\cdot G(x_{0})\cdot Y)$

using the product rule for determinants and $\det(Y^{T})=\det(Y)$. By the
lemma, if $X=(x_{0},x_{1},\ldots,x_{n})$ is a thin $n$-simplex, the $G(x_{i})$
that enter in the expressions for the square distances between vertices in $X$
may all be replaced by $G(x_{0})$. We have by positive definiteness
$G(x_{0})=H^{T}\cdot H$ for some square matrix $H$, and then the equation
continues

$=\frac{1}{n!^{2}}\det(Y^{T}\cdot H^{T}\cdot H\cdot
Y)=\frac{1}{n!^{2}}\det((H\cdot Y)^{T}\cdot(H\cdot Y))=vol^{2}_{I}(H\cdot
Y)=vol^{2}_{g}(Y),$

the two last equality signs by (3) and Proposition 3.2, respectively.
Restricting the $\overline{\omega}$ to the set of 1-infinitesimal
$n$-simplices gives therefore an $n$-form $\omega$ such that

$\Sigma^{\omega}=\Sigma_{g}(=vol^{2}_{g});$

such $\omega$ deserves the name of volume form of $g$.

(An argument for the local uniqueness, modulo sign, of such $\omega$ is not
presented here, but can presumably be given.)

##  4 Second order i-structure

In his Cambridge thesis [1], F. Bár introduced what he calls “higher order
infinitesimal structures” (i-structure) in a manifold $M$, with particular
emphasis on second-order such structures. The crux (for this case) is that
there is a refinement of the notion of 2-infinitesimal $n$-simplex
$(x_{0},\ldots,x_{n})$, where one not only requires that $x_{i}\sim_{2}x_{j}$
for $i,j=0,1,\ldots,n$, but furthermore requires, for all
$i,j,k=0,1,\ldots,n$, that the 2-simplex $(x_{i},x_{j},x_{k})$ is a thin
2-simplex, in a sense which we shall describe now (Bár does not use the word
“thin” though). He first describes the thinness notion for the case where $M$
is a (finite dimensional) KL vector space $V$. He defines $DN_{2}(V)\subseteq
D_{2}(V)\times D_{2}(V)\times D_{2}(V)$ as the set of $(v_{1},v_{2},v_{3})$
such that any trilinear $V\times V\times V\to R$ annihilates
$(v_{1},v_{2},v_{3})$. Then the 2-simplex $(x_{0},x_{1},x_{2})$ in $V$ is thin
if $(x_{1}-x_{0},x_{2}-x_{0},x_{2}-x_{1})\in DN_{2}(V)$. He proves that the
notion of thin 2-infinitesimal 2-simplex is preserved by (smooth) maps, and
hence gives a well defined notion of thinness of 2-infinitesimal simplices in
an arbitrary manifold $M$, (preserved by maps between manifolds): a
2-infinitesimal $k$-simplex in $M$ is then called thin if all its face
2-simplices are thin.

To illustrate the notion of thinness and to show its naturality:

Consider 2-simplices on the line $R$, with first vertex $0$, so consider
$(0,x,y)$. It is 2-infinitesimal if $x^{3}=0$, $y^{3}=0$ and $(x-y)^{3}=0$.
Binomial expansion of the last condition gives
$x^{3}-3x^{2}y+3xy^{2}-y^{3}=0$, and using $x^{3}=y^{3}=0$, this is quivalent
to $x^{2}y=xy^{2}$. But it does not imply that $x^{2}y=0$ or $xy^{2}=0$. This
will hold if the simplex is thin.

Given a simplex $(x,y,z)$ with $x\sim_{1}y$ and $x\sim_{1}z$. Then
$y\sim_{2}z$, so the simplex is 2-infinitesimal. In fact, such simplex is
thin. For, in any coordinate patch from a vector space $U$, we may form the
differences $u:=(x-y)\in D_{1}(V)$, $v:=(x-z)\in D_{1}(V)$ and $w:=(y-z)\in
D_{2}(V)$; so $w=v-u$. Let $\phi:V^{3}\to R$ be trilinear. Then
$\phi(u,v,w)=\phi(u,v,v)-\phi(u,v,u)$, and both terms here are 0, because they
contain $v$, respectively $u$, in a bilinear postion. 2-simplices of this kind
were studied in [4] and [5] under the name “2-whisker” - an (un-symmetric)
weakening of the notion of 1-infinitesimal 2-simplex. The notion of thin
(2-infinitesimal) 2-simplex is a weakening of the whisker notion, and is
symmetric.

## Appendix: The symmetric algebra

Let $k$ be a commutative ring; in the following, “algebra”, “module” etc.
refer to (commutative) $k$-algebras , -modules etc. Every algebra has an
underlying module. The forgetful functor from the category of algebras to
modules has a left adjoint, formation of symmetric algebras. Recall that if
$U$ is a module, the symmetric algebra $S^{\bullet}(U)$ is the value of this
left adjoint, i.e. the free commutative algebra generated by $U$.

It is a graded algebra: in degree $0$, it has $k$, in degree $1$, it has $U$
(and the inclusion of $U$ as degree 1-part has the universal property making
it the unit for the adjunction); in degree 2, it has $(U\otimes U)/2!$, etc.,
so elements in degree 2 are represented by linear combinations of expressions
$e\otimes e^{\prime}$ with $e$ and $e^{\prime}$ in $U$. And $e\otimes
e^{\prime}$ is identified with $e^{\prime}\otimes e$; $e\cdot
e^{\prime}:=e\otimes e^{\prime}$, etc. In degree $r$, it has
$(U\otimes\ldots\otimes U)/r!$, the $r$th symmetric tensor power.

We shall mainly be interested in the case where $U=V^{*}$, where $V$ is a
$k$-module isomorphic to $k^{n}$. So the elements of degree 1 form the
$k$-module of linear maps $V\to k$. If $e_{1},\ldots,e_{n}$ is a basis for $V$
with dual basis $x_{1},\ldots,x_{n}$, $S^{\bullet}(V^{*})$ may be identified
with the algebra of polynomial $k$-valued functions in $n$ vaiables
$x_{1},\ldots,x_{n}$. However, it is preferable not to coordinatize (i.e.
putting $U=k^{n}$, to avoid double indices later. The “semantics” of
polynomials as polynomial functions $V\to k$ is then built in from the outset,
since the module of degree 1 elements is taken to be $V^{*}$, thus its
elements are (linear) functions $V\to k$.

So we consider the graded algebra $S^{\bullet}(U)$ for a general $k$-module
$U$, with $S^{1}(U)=U$, $S^{2}(U)=(U\otimes U)/2!$, …

$S^{\bullet}(U)=k\oplus S^{1}(U)\oplus S^{2}(U)\oplus\ldots$

It has some particular ideals, $I\supseteq I_{2}\supseteq
I_{3}\supseteq\ldots$ where $I$ is the ideal generated by the degree 1 part
$S^{1}(U)$, $I_{2}$ ($=I^{2}$) is the ideal generated by the degree 2 part
$S^{2}(U)$, etc.; equivalently, $I_{r}=\bigoplus_{j=r}^{\infty}S^{j}(U)$.

Next, we consider the $k$-fold tensor product ($k$ denotes here a natural
number, not the ground ring $k$) $(S^{\bullet})^{k}(U)$, meaning

$S^{\bullet}(U)\otimes\ldots\otimes S^{\bullet}(U)\cong
S^{\bullet}(U\oplus\ldots\oplus U)$

($k$-fold $\otimes$ and $\oplus$). The left hand side here is multi-graded
($k$-graded); The elements of multidegree $(r_{1},\ldots,r_{k})$ form the
direct summand $S^{r_{1}}\otimes\ldots\otimes S^{r_{k}}$. Also the symmetric
group $k!$ evidently acts on $(S^{\bullet})^{k}(U)$, so we may consider the
subalgebra of $k!$-symmetric elements; but we may also consider the submodule
og alternating elements. (In particular, alternating elements of multidegree
$(1,\ldots,1)$ in $S^{\bullet})^{k}(V^{*})$ correspond to $k$-linear
alternating functions $V^{k}\to k$.)

The following is now a simple “counting degrees” argument: if
$\omega\in(S^{\bullet})^{k}(U)$ is of multidegree $(1,\ldots,1)$, and if
$\theta\in(S^{\bullet})^{k}(U)$ is of multidegree $\geq(1,\ldots,1)$ and of
total degree $\geq k+1$, then in the algebra $(S^{\bullet})^{k}(U)$ we have
that $(\omega+\theta)^{2}-\omega^{2}$ is of total degree $\geq 2k+1$. For, the
difference is $\theta^{2}-2\theta\cdot\omega$. Here, the first term is of
total degree $\geq 2(k+1)\geq 2k+1$. In the second term, $\theta$ is a linear
combination of terms of multidegree of the form $(1,1,\ldots,1+p,\ldots 1)$
for $p\geq 1$, so $\theta\cdot\omega$ is a linear combination of terms of
multidegree

$(1,1,\ldots,1+p,\ldots,1)+(1,1,\ldots,1,\ldots,1)=(2,2,\ldots,2+p,\ldots,2)$

which is of total degree $2k+p\geq 2k+1$.

## References

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