So that will speak about sums of course
but we will begin a long time ago

in a galaxy far far away
there was a graph OK so

I'm going to index the Verde season
my graph I just want to end and

a subset of the graph to the stable set.

Also sometimes called
an independent set if no two.

In S. are connected by an edge of the
graph so you see here a picture of a bunch

of stable sets in the graph.

And now well so what I want to do
is I want to study a stable set

geometrically so instead of just having
a subset of every stable set I'm

going to associate a characteristic vector
so it's going to be that there in R.

N. and it's the number of how am
I going to index the stables and

I'm just going to put one if
the vertex is in the stable set and

I want to zero if the vertex
is not in a stable set.

OK So was this now what can I do now I
can form of politics is just the convex

hollow all the characteristic of actors or
wall the stable sets and the grab G.

and I call it step G. so that you want
sort of the important thing to know is

that I don't take only maximal stable sets
I really take all of the stable sets So

for example zero is always going to be
stable Set politics because empty set is

trivially stable OK So this is
the definition of the stable set politics.

So well first well there are two
questions that you might have.

One is why would I do such a thing so
this is kind of a standard construction in

common to optimisation if
you want to let's say find

the maximal size stable set or if you
have weights and roots you might want

to find a maximal weighted stable set
well so if you have a good description.

Of this fall then the century
maximizing say finding a stable set of

maximal size is just maximizing a linear
function which is the sum of excise or

this politics that is going to
maximize at a vertex this vertex is

going to give you a stable set of
maximal size OK So the thing is

that what we need is a good description
of a stable set polygon of course I gave

you the description I defined of this
the conduct solvable by introverted C.S.

but this is not good this is so far you
know it does not really contribute very

much for me it's just like saying that I
have all of the stable sets so what we

really need is a good description and so
what we need is the dual description so

the dual description means that I
want to describe the stable said

not by brute force used but
by inequalities OK So I mean for

a new policy all dispositive also the
question is how to get the inequality so

let me give you some inequalities
that are trivially true for

the stable set follow so
because it's in that stable said by

characteristic that there is this
means that any point in the politics

has to have a non non negative coordinate
right all coordinates are not negative and

all coordinates how less than or
equal to one so this is obvious but

of course this just defines for
me the cube right but

I really want to define the stable set
follow it up so how how do I construct

more inequalities that are valid
on the stable set Poly to.

It so I want more linear inequalities that
are valid on the stable set volatile but

of course if I have a linear inequality
that is valid on the politics with some

factors not for me to be valid
on all of the virtue C's OK So

let me invite the set of all of the roots
of C's of the stable set volatile so

T. is just a set of all
the characteristic vectors and

the set of all the characteristic
of all of the stable sets and

now OK now we have this is
the second question How why is this

all related to sums of squares So
where does this come.

So what I want to use sky for
you is an algebraic method.

To generate linear inequalities
that are valid and

I guess this was supposed to be so
basically what I want to just have to use

an algebraic method to generate
inequalities that are valid on this finite

set of points this is where sounds of
course coming OK So so the question for

us is how to generate linear inequalities
that are valid on this finite set and

if I generate enough then perhaps I will
really cut out the polytope exactly how do

we generated So that's were sums of
course common So let me construct let me

try to construct linear functionals that
are obviously going to be not negative.

Now what do I mean by obviously negative
well so you know if I give you a linear

function you might wonder I mean how do I
know that it is not negative OK so low so

basically what I will what I'm trying to
build up functions that are going to be so

negative on T. that there is not going
to be any question So what do I mean so

let's imagine that I give
you a linear functional and

I can write it algebraically as
a sum of squares of polynomials

Plus another polynomial that is ear on C..

OK then I claim that this linear function
if I can do it this linear function is

obviously a negative one T. Y. is that
while because some of the squares are not

negative everywhere that's that's a good
thing there's no question that sounds

of course are negative for now if I add
to it Apollo normal that is zero on T.

that does not perturbed its values on T.
right so therefore

this means that some of course is not
negative so it's not a negative one and

now I add something that zero on C.
so it doesn't affect me at all right so

this means that these guys are in fact you
know if I can write a functional like this

then L. is clearly non-negative on C..

So but of course the thing is that I'm
not really interested in very conflict.

It is so you could see you could think of
this as you could think of this expression

as being a certificate
of negativity on T.V.

If I give you that if I give you an
expression like this and it's provides for

me a certificate now I know that
the function has no negative C. but.

For computational reasons I'm not
really interested in the extremely long

certificates I mean because this is
a finite set you can sort of prove

that these expressions exist in general
right in your functional if you for

the negative can be certified like this
but what you want to look at are functions

that we can compute were ordered
certificates that we can compute And so

what this means that what we want to do
is you want to try and create the degree

of the scale of the problem normals
that we are allowed to square OK so

what we're going to do is we're going to
loop at linear functionals that can be

written as a sum of squares so
fall enormous of degree at most K.

plus something that is zero on C. so I'm
going to try and get the degree so I'm not

only allowed to square simple things and
then I can add to it something from C. and

if I can somehow cancel a bunch of things
to get a linear function I know that those

are the functions that I'm interested
in OK and such a linear function so

I'm going to call them K S O S S O S
stands for sums of squares case stands for

the truncated degree OK So these
are the function of them interested in so

these are let me just say that maybe in
order to ease our linear function is that

I can write as a sum of squares of simple
things plot something that is zero on C.

OK.

Are there questions right

now.

So we will get so we haven't gotten
through the to the bottom of a slight I

will get to the bottom of
the slide in in a second.

And your other questions yeah.

I well so the thing is that your case so
sums of squares you know if I if I

add the mob that will have high degree by
I have the freedom to add to it something

that is zero on T.
so that can cancel things out it might

not seem that might seem like strange
phenomena doesn't fact happen if I add

things that are your own to you that can
cancel all of the terms of high degree.

OK.

We're OK great so
the of course then you know the next

thing that should to wonder is well when
I say I add something that is zero on T.

is that easy to describe in a what
are the things that are zero in T.

because if I'm adding something that's
potentially very complicated then

you know I mean this looks
nice in principle but

what are things that are zero and
see if they're very complicated and

I'm not really sure what I'm cancelling my
higher degree terms with so in this case

because we're working with this comes from
a comment or optimisation problem fact

it's very easy to describe all
polynomials that vanish or

so they are generated by these guys so
the ideal this or

that vanishing T.
has very simple generators so

one is I mean because so
far we decided to index.

Index the stable sets these are zero one
that So this means that you know for

every coordinate they have X. a square it
is equal to excite that just means that

we're talking about zero one doctors so
now what are the other equations so

the other equations are so whenever
I N J Were number wrote excise connected

to vertex J by an edge I throw in X. side.

Now while why is that it's because
you know if I have to work to see is

that are connected by an edge both of
them cannot be in the stable set right so

this means that one of the coordinates
one of them was not present so

one of the coordinates is going to be
equal to zero so going to vanish and

in fact these are all.

All of the everything else is
generated by these guys so

this is a very nice description of
this idea so it's actually easier for

me to say what our problem normal is
there are going to do than Ashanti OK So

but so this is the plan so this is kind of
my plan of attack how do I generate lean

your inequalities that are valid
on the whole on this finite set.

Of.
Stable set of index stable sets Well I try

to write the linear inequality as a sum
of squares of degree at most K. and

then add to it something that is zero in
C. and hopefully I can cancel I can cancel

higher degree terms and
truly get a linear functional OK.

All rates.

So what does this do so
now let's look let's look instead at.

Sequence OK so I have the set of an
equality that comes from sums of squares

of degree at most K So let's look instead
at the convex bodies that they describe So

this all of these inequalities are going
to describe some convex bodies so

I'm going to call th K.
so like the key theatre body is the set of

points in our in which are actually
cut out by the linear functionals that

comes from some discourse of degree
at most K. so this provides for

me a sequence of relaxations
all of my original

stable set stable set volatile sits inside
all of them but of course you know about

these guys are nested so the first line
you know constant sums of scores of Lynn

year is going to be potentially bigger
than the second one potentially bigger and

then they nest national until I should
get to the stable set volatile and

in fact this is going to
terminate finite me at some point

at some point I'm going to fight
big degree high enough I'm actually

going to pick up all of the facets
of the stable said politics but

of course I mean if I mean well OK
fine I think it's not really us.

Interesting what we want to know
is when does this terminate for

small values so bam OK So and

then there is a theorem of Lovaza So
this tells you what happens.

When the when does this when does this
relaxation in fact terminate in the first

step so the first realisation So the first
thing that a body is equal to exactly

the stable Set politics if and
only if the graph is perfect so

this is kind of interesting but.

The behavior of these guys is not very
well understood so if I ask you for

what graph G. doesn't happen then
the second you know now is for

the second is equal to the stable set
policy of this is completely completely

open so the reason is that somehow well
when you look at when you look at this I

mean you can really define you can really
define disperse relaxation not just for

a stable set Pollard or for a new politics
and somehow to have the first relaxation

being equal to the politics of this
is a very restrictive condition

you can pretty quickly say you know for
which probably does the but

which probably does the sap and of course
then you later need to realise for

which graphs does this happen but when you
ask for which volatile does it happen that

the second relaxation is exactly equal
to the power then somehow this is not

restricted were enough to be able
to say much at least so far.

And let's see so basic I mean for
me I just want to bring the sin

as a motivating example so another thing
that I just wanted to add of the wards.

Feta function mean something to you
that right now would have seen the first

relaxation really is like the lower state
of function in disguise even though most

of the time when you see the definition
of the law of our state a function

you do not talk about
sums of squares OK so so

somehow this is discussed yet.

That.

So this is the this is coming up right so
all right so

basically I just wanted to make this
stable set as a motivating example so

here are so here are sort of my mean
points that I want to illustrate

by this example so
what what exactly were we doing.

So basically we wanted to study all
non-negative linear functional so

in a certain set right but
what we did instead we relax and

we said well we cannot really study
all non-negative guys about lead just

only study things that are sums of
squares write sums of course and then for

computation reasons we truncate it by
degree Now the reason why is the same you

good right why I mean well some of
course intuitively it's clear why this

is good because some of the squares are
obviously in a negative but this doesn't

mean that it has to be computationally
good right so the reason this is good is

because some of the squares can really be
handled by semi definite programming so

what we're doing is really like the law
of our state of function is really

semi definite relaxation that's
how it's usually gets defined so

what we're really doing is constructing
some specific semi definite relaxation.

And somehow there are a lot
of open questions so

I already mentioned like what happens
when the second one does the second

relaxation really hit the stable set
follow it so you know you can ask it for

witchcraft does this happen before we
stable Seppala dog so you can just ask for

which followed you can define
the same relaxation for

and you probably don't doesn't really
have to be the stable set one.

And this is.

Completely completely open and the
difficulty really comes from the fact that

we truncate by degree if we don't try and
get by degree then it's pretty easy and

you know even if you have more complicated
sets doesn't have to be you know just

finite set of sets of points there
are some powerful tools from realize

geometry that tell you that
good things you know once

you lead the degree around to infinity
good things are bound to happen.

But that's not really
sort of interesting for

us you want to know what happens
when we truncate the degree.

New generally want to try and
get the degree to be a pretty low this

gives a sort of good relaxations right and
then then somehow these tools which had

their your powerful and
very general do not apply and.

So then this is kind of the difficulty

of the analysis then the final thing the
kind of is lower state of function that

often familiar semi definite relaxations
are actually sums of squares

relaxation in disguise I mean you do
not have to think of them or some.

But they happen to be some of the squares
if I if you look at them in the right way

they really come from sums of squares
like pretty sure that the government's

Williamson max card relaxation you could
look at as sums of squares relaxation

Also if you if you are familiar with work
or Frank Valentijn on kissing number so

spears which is really like independent
numbers of infinite graphs so stable set

number of infinite graphs also in that
case you can think about it as a sums of

squares relaxation so somehow sums of
squares are stationed there specific semi

definite relaxations but they happen
to occur fairly naturally OK And

this is I mean this is what so
do so me I just want to motivate

using sums of squares as a way
to handle negativity so.

I mean I also work on
common Atauro questions but

not really sort of exclusively
common authorial questions so

we're about to we're about
to shift gears a little bit.

So are there questions right now.

We're good all right so.

We're shifting gears OK so now I'm just
going to think about pollen normals

that are not now negative
on a certain set but

I'm going to think about polynomials that
are not negative everywhere right so

polynomials that are really
global in the negative.

And of course I can still think
about sums of squares but

now when when things are negative.

Some some phenomena that we have observed
cannot happen for example there is no

degree cancellation you know there is no
way that I can write a polynomial of.

Degree.
To Dia sums of squares of things of higher

degree and another thing that happens
if a polynomial is not negative

then we can make it home with genius
by adding an extra variable and

then multiplying all of the terms by
an appropriate power of the extra be

able to make all of them
anonymous have the same degree and

this actually from Paul normal was not
negative it will keep being negative

right then if the polynomial was a sum of
squares that it will keep being a sum of

squares So what does this mean so
the bottom line is that if I

want to study globally in a negative fall
normals I might as well study globally

non-negative how much genius problem is
also called forms in one more variable so

that's what we're going to do we're just
From now on if I say probably normal

i will mean a homage genius polynomial and
now OK so now let me fix so

and it's going to be a number of variables
two D. is going to be the degree so.

All fall in normal sin this man U.
they're both variables of degree two D..

Form a vector space form
a large vector space

inside this vector space see it in the sit
two sets that are we're interested

me the set of non-negative polynomials and
the set of sums of squares but

somehow an interesting things happen these
are not just sets in fact they are fool

dimensional close convex cones it's
pretty easy to see why they're convex or

so for example what does it take to
be a convex going this means that if

I take two points in the cone and I add
them the sum was still in the cone and

if I take two non-negative
problem normals and

I add them it's still not negative right
so it's pretty clear that these are two

candidates goes it's also pretty
easy to show that they're closed.

OK so now what is the first thing
that you would ask is Well when a.

Two things equal right when at least
two candidates crones equal so

the whole subject of study of course and

the negative almost were started
by a theorem of Hilbert.

We should say she qualified classified all
of the cases where there is equality so

there is a quality only in the following
sort of three families of cases

case one and equal studio two very robust
better remember this is two variables how

much genius The same is one variable not
homogeneous it's pretty easy to show that

any non-negative problem normal and
one variable is a sum of squares case two

to D. Of course two this is
the case of chord radix forms so

that's pretty easy to do by
diagonalization of course forms and

Case three is an equal three to four so
so called two hundred cortex and in

all other cases in all other cases there
exist no negative forms that are not sums

of squares and so it's kind of interesting
Hubert's proof was non-constructive But

later in know by now there are several
papers some papers so resonate

there particularly nicely provide some
explicit examples in explicit family so

the first example was due to Matson but

now there are sort of meanings out what
somehow it still it still seems hard

it still seems hard to construct
explicit examples of non-negative

polynomials that are not sums of squares
so you could ask well maybe OK So

these are two convex cones right but
maybe they're close to each other

maybe you know this is go this is kind of
a mental picture you know here is some of

us who are as they sit inside the negative
cone force and not two dimensional but

maybe these two sets are really close
to each other that's why we really

have problems picking off things
you know that are now in one but

not in the other but
because we have convex set so

we can actually try to address this thing
by measuring their size well located cones

are on bounded objects right we cannot
really measure the size of the cones so

what can we do we.

And slice it was a hyper plane
because slice it was a hyper place so

in fact we have to compact objects.

So we have to compact objects so
once we have two compact objects

we can take volumes right so
we can really take volumes so

now by there is a little bit of a carry
out so if I take a compact set and I blow

it out by a factor one plus Sept so on
the volume is going to be multiplied by Y.

plus absolute want ourselves
on to the ambient dimension so

if my Ambien dimension is large then I'm
still in trouble because even if I blow it

out by a small factor the volume is really
going to get blown out of proportion so

what we need to do is we need to take the
volume and raise it to the power one over

Ambien Dimension This takes
care of this effect OK So

OK And so
now we have sort of bad news that says So

let's take this ratio volume of the slice
of the non-negative problem most divided

by the volume of a slice of sums of
squares raise it to the power of one over

dimension how does it behave so we're
going to be in the regime where we keep

the degree the same so for example we talk
about all polynomials of degree four and

we'll let number of their books run to
infinity then we can actually get the sort

of exact awesome thought ICS
exact awesome fabrics of this.

Of the of this behavior so well so
what happens why is the news bad so

let's look at the case when
the total degrees for so

the total degree from east to D.D.
So two D. is forced to do so

then already it says that this
normalized ratio values behaves like and

to the power of one half right so
this is already bad so

this means that a convex sets the set
of non-negative problem Norma's is

really much much bigger than
the set of sums of squares.

OK but there is there is some actually is
somewhat mysterious things that happen

OK so mysterious and well not really
mystery in one but you know I swept a lot

of things under the rug by saying
that there are these some constancy.

One and see two these constants are in
fact as usual extremely ugly so

what it means is that the ratio is
only guaranteed to be very large for

a very large value so event so for
a larger Savan the value so when so

large that you might say you
don't really care about well

as from the point of view of
complexity of course you do but.

Basically if I cannot guarantee that this
ratio is above one I mean using this bound

for the way when N. is let's say a billion
then you know maybe there is something

something wrong here and.

Listen thing that you might ask yourself
is Do I really do I really really care

about random polynomials I mean if I
take a polynomial or something like

That's a degree and a thousand variables
that's going to have a lot of Manami else

it's going to have so many more normals
that will never want to write it down.

So this means that basically what you well
if you really want to write things down

and work with something in many variables
they better have some structure so

really you might not really ever wonder
about the random but you might want to

know about something that has some
structure so this brings me to the.

Last flight of the talk so

if you can force a lot of
structure then there is some hope.

For what happened so I'm going to
really enforce a lot of structure

on on my problem or some going to do when
you look at symmetric polynomials so

this means polynomials that are going
to be invariant with respect to

any permutation of variables there so
if I look at symmetric followers and

I look at the ratio volumes I don't even
need to normalize it by the one over

him in the mention this that is
not grow infinitely large for

any degree D. for any well and
you need to D. and in fact for

degree for this ratio I actually go
through one so when I look at symmetric

non-negative problem normals and I look at
symmetric sums of squares as engrossed.

The infinity of the situations on how
get so well it's not monotonic but

it gets better and better and
you get some more and

more of not negative problem normals So
this is good news and

another good news i even
if you want to analyze.

It and even if you want to analyze let's
say all non-negative follow Mills You

could try to look at what happens for
small cases where the cones are not

the same so the two smaller skaters
are three variables degree six and

four variables degree four because that's
just outside of the case where there is

a quality during record ticks and
in this case we have in some sense a full

description of when the qualities
that separates us from the negative

follow males but in his own right now it's
not so clear what to do with this will

description how to sort of piece it
together in the overall analysis but

there is a hope of some understanding of
what happens in the negative problems and

some of course and especially it's
interesting I mean what happens

in the present I mean this is really a lot
of structure symmetric problem of several

I'm really cutting down my freedom
by a lot when I was six or

symmetric problem on the so what happens
when I require less structure OK So

this is all that I wanted to say so
thank you very much for your attention.

I.

Know actually if you allow rational
functions then in fact any non-negative

problem on THIS WAS Hilbert seventeenth
problem any non-negative polynomial is

a sum of squares of rational functions.

That's right that's right it's
the same the problem is that it's hard

to know what to use as denominators So
there is sort of there that's good that's

it somewhat to cool and
to a degree truncation in

a certain sense it's hard to know how do
you use denominators of limited degree.

So essentially So
the there is the say in the So

again this is I would say the flavor
of the idea is the same there is

a very powerful result from realize that
says that good things are going to happen

if you allow yourself
sort of arbitrary but

then you ask well what happens if I
don't allow myself arbitrary degree but

what happens if I if I only allow
myself denominators of small degree and

then somehow it gets very murky
very difficult to analyze and

so there is not much known.

What happens when the degree of
the numb there is not very large.

And.

Then OK so it's they in fact it's not it's
not in a sense it's not really an answer

it were it's it's it's actually
it's a strengthening of the actual

some of the squarest relax a you can
really you can model this with some of

those who are so large a degree so in
a sense it's so let me put it this way so

usually usually the result Well one way
to state the result is the following so

if I have a negative polynomial P. O. X.

I can multiply it by a sum of scores and
in the end I will have a sum of

scores this is this is equivalent to being
a sum of course of rational functions so

the only question is how to find this
multiplier of small degree but you can

still state it as a semi definite program
you could say for given a fix problem

normal does there exist a multiplier of
something degree that makes it a sum of

squares this will still be some indefinite
relaxation so you can so you can say that

this you can strengthen the sum of
squares relaxation by asking for

some of the squares with
a certain multiplier OK and

this will give you a stronger
semi definite relaxation.

Yeah.

That's right so I mean essentially OK let
me and this is this is the arrow also

brings us to something General from
algebra so basically what is special about

the entire world zero infinity is defined
by X. create in the requote two zero

right so this means that really what are
things that are obviously not negative for

this well sums of squares right this
is obviously a negative I can also

multiply by X. right this is going to
be obviously in a negative right so and

now the question is when
does it happen that.

Basically you know I can just throw
in more inequalities right so I have

something that's defined by I have a semi
algebraic said defined by inequalities and

I ask is it true that every negative
problem normal can be written as sums of

squares sent potentially multiplied
by someone equalities OK end so

depending on so depending on your
assumptions so for example if your set

this compact then the answer is true
always you know in many dimensions

in variables if your set is not compact
this happens to be true sometimes or

it often happens to be true if kind
of like here you allow multipliers So

if you allow multipliers if you say not
and I don't want to write my negative

problem of the sum of squares about allow
someone who has multipliers then this is

always true and all of the serum
sort of by the name of positive

it's kind of like no shell and such except
you know positive because we're talking

about in The Call it is opposed to zeros
and then you attach a certain name to it

there is shingles positives Jones outs and
was general.

But is so you attach names to sort of
apply to a specific situation what kind of

said Do you have and so there are certain
flavors of theorems that tell you that but

generally this is something like
this is true if you just mom or

if you modify your statement enough
like if you allow multiplies or

if your set is compact
this will always be true.

We need to.

Thank you.