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Fixed point of gradient flow for polynomials with positive coefficients.  A Nice Theorem Wanted!
Since 1998/08/25
update 1998/10/18
(Conjecture 2 and Result for W with degree no more than 4)
update 1998/10/31 (Counterexample to Conjecture 1,2)
update 1998/12/27 (Results for W without terms linear in y)
update 1999/02/04 (Formulation for W with terms linear in y)
update 2006/10/06 (Results on uniqueness)
I have been trying to find a Theorem, since early 1990s,
which should state, under some 'nice' conditions,
existence of a unique critical point for a polynomial with two variables.
The road I took was winding, but results are steadily accumulating,
and I am absolutely certain that there are nice Theorems for this problem.
I greatly appreciate any suggestions!!
Problem (19981031).
Let
D={(x,y) in R_{+}^{2}  {(0,0)} 
y <= x^{2} }
be a subset of the first quadrant R_{+}^{2} in the
two dimensional plain, and
let W: R_{+}^{2} > R be
a polynomial with two variables, with the following properties:

W is a nonempty sum of terms each of total degree 3 or higher and with
positive coefficients.

grad W(D) is a subset of
D  {(x,y) in D  y = x^{2} }.
Here,
grad W=(W_{x},W_{y}): R_{+}^{2}
> R_{+}^{2}
is the gradient map of W. (W_{x} is a partial
derivative of W by x and W_{y} is that by y.)

W(x,y) contains a term linear in y (a term of a form x^{n}y
for some n>= 2). In other words, Y(x,0)=W_{y}(x,0) is not
identically 0.
Find sufficiently general, easy to check, and natural, sufficient conditions
with which the critical point of the
polynomial w(x,y) := W(x,y)(x^{2}+y^{2})/2
defined by grad w(x,y)=0, uniquely exists in D.
Remarks.

Critical point of the polynomial w=W(x^{2}+y^{2})/2 is
equivalent to fixed point of the gradient map grad W.
The second assumption on W implies that D is an invariant domain of
grad W.
(The assumption is stronger in that the parabolla y=x^{2} is not
in the range of the map. This is to focus on 'generic cases' by avoiding
possible cumbersome considerations
on the 'boundary' or 'when the equality holds'.)
Click here for a motivation for chosing
D as an invariant domain.

How general should 'sufficiently general' be?
The original motivation was on the two explicit polynomials
at the end of this page. Therefore, at least these examples should be
contained in a 'natural' way.

The case when W(x,y) has no terms linear in y
should be treated differently from the case when a term linear in y exists.
Then there is a unique critical point of w (fixed point of grad W) on the
positive x axis, and Problem (19981031) is equivalent to finding
sufficient conditions such that there exists no critical point of w in
D  {(x,0)x>0}.
For details, click here.
The more interesting case is for W(x,y) with linear terms in y.
We therefore added the 3rd condition in Problem (19981031).
Then there is no critical point of w (fixed point of grad W) on the
positive x axis, and Problem (19981031) is equivalent to finding
sufficient conditions such that there exists a unique critical point of w in
the interior of D.
Existence of critical point.
(1) Approach from topological fixed point theorems.
We can prove (using the methods in K.Hattori, T.Hattori, S.Kusuoka,
Probability theory and related fields 84 (1990) 126) that
under the assumptions in Problem (19981031),
there exists a critical point
of w in the interior of D.
The proof uses the asymptotics of orbits of the dynamical system
generated by grad W.
To be more precise, that W is a polynomial with positive coefficients
implies that D is the union of three disjiont invariant subsets of the map
grad W, which are the set of points whose trajectory

converge to the origin,

diverge to the infinity,

stays bounded away from the origin and the infinity.
Then the fixed point theorem implies the existence of a fixed point
in the third set.
Note that, in using topological fixed point theorems as in this approach,
decomposition of the set D by the
invariant sets of grad W is essential,
because topological fixed point theorems allows fixed points at the
boundaries of D, whereas, in our problem the origin 0, which is in the
boundary of D, is trivially a fixed point of grad W.
Notes added (20061006).
The reference above corresponds to the problem for W with 1 variable.
The following references deal with the problem which corresponds to
the problem here for W(x,y) with 2 variables.
(The idea used there is as explained above.
If you read Japanese, I also wrote a book
'Random walk and renormalization group'
(Tetsuya Hattori, 2004, Kyoritsu publ.), which integrates all the references
and systematically explains the problem.)

K. Hattori, T. Hattori, Sh. Kusuoka, Selfavoiding paths on the three dimensional Sierpinski gasket, Publications of RIMS 29 (1993) 455509

T. Hattori, T. Tsuda, Renormalization group analysis of the selfavoiding paths on the ddimensional Sierpinski gaskets, Journal of Statistical Physics 109 (2002) 3966.
(2) Approach from inverse function theorem.
By adding rather weak assumptions stating onedimensionallike situations,
one can prove the existence of critical point in the interior of D
without using the asymptotics of dynamical system.
Proposition (19990109).
In addition to the assumptions in the Problem (19981031),
assume the following
properties for (X,Y)=grad W:

R(x,z)=X^{2}(x,x^{2}z)Y(x,x^{2}z) is a polynomial in
z, 1z, and x, with nonnegative coefficients.

It holds that
R(x,x^{2}z)/Y(x,x^{2}z) = O(x), x>0.
Here, O(x) is meant to be an estimate uniform in 0 <= z <= 1.
Then, there exists a critical poiont of W (a fixed point of grad W) in
the interior of D.
Remarks.

I called the first two conditions onedimensionallike, in the sense that
under these conditions, Y is close to X^{2} uniformly in
0 <= z <= 1, when x is close to 0, hence the orbits of
the dynamical system generated by grad W approach 0 approximately along
a parabolla y=x^{2}, so that the original picture that y is
of higher order compared to x becomes clearer.

For a proof, click here.
Uniqueness of critical point.
Let us turn to the uniqueness problem.
First note that if all the terms of W has ydegree 2 or less,
then the uniqueness is obvious. Click here for
a statement and proof.
This case is too simple to inspire solutions for the general case.
It is not easy to prove the uniqueness of the critical points of w in D,
when W has terms of both ydegree 1 and ydegree 3 or more.
The following are two specific examples of polynomials which
satisfy the assumptions and the conclusions of the Problem (19981031).
These examples are related to the original problem
in the renormalization group for selfavoiding walks on Sierpinksi gaskets.
Proposition (19990109) inspires the following.
Problem (19990204).
Are the assumptions in Proposition (19990109) sufficient for
the uniqueness of the fixed point in the interior of D?
Other comments.
The recorded date implies that more than 7 and a half years has past
between this line and the lines above. Meanwhile, various examples and
experiences convinced me that the Problem (19990204) is essentially
true.
Conjecture (20060819).
In addition to the assumptions in the Proposition (19990109),
assume that W(x,y) contains (i.e., with positive, nonzero, coefficient)
a term x^{3}. Then The critical point of w in D is unique.
We remark that the additional assumption specifies the lowest order term in W.
The case of a function with 1 variable suggests that
that uniqueness of critical points hold true otherwise,
but since we would rather avoid technical difficulties in the proofs,
we shall make life simple.
Even with the simplification, a proof of the Conjecture (20060819)
turned out to be
very difficult, as may be correctly guessed from the 7 and a half years'
absence of updates for this page.
Theorem (20060819).
In addition to the assumptions in Proposition (19990109),
assume the following.

W(x,y) contains a term x^{3}
(i.e., its coefficient is poositive nonzero).

Each term in W(x,y) has a total (x plus y) degree 6 or less.

The terms with positive powers in y has total degrees 5 or more
(i.e., 5 or 6).

W contains no term of the forms xy^{4} or x^{2}y^{3}.
Then there uniquely exists a critical point in D of the polynomial
w(x,y)= W(x,y)(x^{2}+y^{2})/2
(i.e., a fixed point of grad W).
We can make the assumptions of Theorem (20060819) explicit.
Proposition (20060819).
The assumptions of Theorem (20060819) for W are equivalent
to the following.

W(x,y) = a x^{3} + b x^{4} + f_{5} x^{5}
+ f_{6} x^{6} + (3 a x^{2})^{2} y
+ g_{5} x^{5} y + h_{3} x^{3} y^{2}
+ h_{4} x^{4} y^{2}
+ n_{3} x^{3} y^{3}
+ a_{24} x^{2} y^{4} + a_{05} y^{5}
+ a_{15} x y^{5} + a_{06} y^{6},
where the 12 coefficients
a, b, f_{5}, f_{6}, g_{5}, h_{3},
h_{4}, n_{3}, a_{24}, a_{05},
a_{15}, a_{06} are nonnegative, and a>0.

R_{n}, n=5,6,7,8,9,19, defined by the following, are nonnegative.

R_{5} = 24 a b  g_{5}  2 h_{3} ,

R_{6} = 16 b^{2} + 30 a f_{5}  2 h_{4} ,

R_{7} = 216 a^{3} + 40 b f_{5}
+ 36 a f_{6}  3 n_{3} ,

R_{8} = 288 a^{2} b + 25 f_{5}^{2}
+ 48 b f_{6} + 30 a g_{5} + 18 a h_{3}
 5 a_{05}  4 a_{24} ,

R_{9} = 360 a^{2} f_{5}
+ 60 f_{5} f_{6} + 40 b g_{5}
+ 24 b h_{3} + 24 a h_{4}  5 a_{15} ,

R_{10} = 648 a^{4} + 216 a^{2} f_{6}
+ 18 f_{6}^{2} + 25 f_{5} g_{5}
+ 15 f_{5} h_{3} + 16 b h_{4}
+ 9 a n_{3}  3 a_{06} .
W=W_{3} and W=W_{4} a little above Problem (19990204)
are a couple of examples satifying the conditions of
Theorem (20060819) or, equivalently, those of
Proposition (20060819).
Simpler examples are

W(x,y)=W_{u}(x,y) = x_{3}/3 + x_{4}y
+ u y_{6}, 0 <= u <= 8/3.
What is nontrivial?
Note that, restricting the possible terms of W to 12 types as in
Proposition (20060819), the nontrivial nature of the
original problem are not lost, and it is still necessary to
restrict ourselves to the set D, to state uniqueness results.
In fact, for the example W=W_{u} just introduced,

for 0< u < = 8/3,
there are 4 fixed points of grad W in the 1st quadrant;
(0,0), (0.662...+O(u),0.192...+O(u)), (0,(6u)^{1/4}),
(O(u^{1/8}), O(u^{1/4})),

while for u=0, the last 2 fixed points are absent,
and there are 2 fixed points of grad W,
hence the number of fixed points (critical points of w) is unstable
with changes of W, much less any reasonable uniqueness results
outside of the set D.
Furthermore,
Theorem (20060819) implies that the qualitative picture for
small positive u continues to hold up to O(1) values u <= 8/3.
That there is an upper limit in u is not the weakness of our Theorem,
because for u around 10, the 2nd fixed point enters D, and at the value
of u little larger than 18.3488, the 2 fixed points meets and vanish
for larger u.
Stability of unique existence of fixed point in D is true only for
bounded range of u; it is a highly nontrivial phenomena.
How come the conditions?
The uniqueness results were long known to hold for the examples
W=W_{3} and W=W_{4}, for which grad W are
essentially the renormalization group for
the restricted selfavoiding walks on 3 and 4 dimensional gaskets.
Though the Conjecture (20060819) seems hard to prove,
it seems reasonable to expect that there are good class containing
W_{3} and W_{4} such that the uniqueness results hold.
The conditions in the Theorem (20060819) are
found as properties which are easily derived (without e.g., counting
the number of selfavoiding paths) in the background problems for
W_{3} and W_{4}.
Though the proof of Theorem (20060819) which we have does not use
any intuition from the properties of selfavoiding paths,
by yettobeunderstood coincidence, the terms avoided by the conditions
in Theorem (20060819) are bad terms for our proof.
I will leave the precise relation between the polynomial W and
the problem of restricted selfavoiding paths on the 3 and 4 dimensional
gaskets to the references above. If you read Japanese,
there is a detailed and systematic explanation in my book
'Random walk and renormalization group'
(Tetsuya Hattori, 2004, Kyoritsu publishing).
It is perhaps worthwhile pointing out that the conditions that
the coefficients of W are nonnegative and that their combinations
R_{n} are nonnegative, are, in the examples
W_{3} and W_{4}, derived from the nonnegativity of
path numbers (which convey to nonnegativity of probability measures)
and selfavoiding properties, and are essentially conditions based on
inequalities.
They correspond to thermodynamical stabilities, hence are naturally
inequality type of conditions.
It may be that there are natural classes defined by inequalities,
for which fixed point theorems hold.
How was the proof found?
There seems to be little mathematical results which work for
our problem. The proof I have is by brute force and trial and error.
I wrote the paper
to which I refer for a proof.
Click here for a look into the crucial calculations
which may show how brute the proof is!
What are the open problems?
There are a lot.

The proof I have is by brute force and very strong.
It gives stronger results than the stated Theorem,
but are too strong to be applicable for all the W in the class
assumed in the Conjecture (20060819).
It is a totally different problem to find an alternative proof
applicable to wider class.

Extensions to W with 3 or more variables are of course of interest.
In fact, the background problem of the restricted selfavoiding paths
on the d dimensional gakset corresponds to W with [(d+1)/2] variables.

Existence of fixed points leads to continuum limit construction of
continuous time selfsimilar
processes with nontrivial fine structures (e.g., Haussdorff dimension
greater than 1). Meanwhile, to prove asymptotic properties
(e.g., displacement exponents and laws of iterated logarithms)
for selfavoiding paths (with discrete unit steps),
we need to prove the existence of convergent trajectory of
the dynamical system determined by grad W to a fixed point in D.
The class in the Conjecture (20060819) and
Theorem (20060819) contains examples
for which the convergences of the trajectories are oscillating.
Therefore we might need another natural class, smaller than those
considered so far, for the convergent trajectory problems.
References

T. Hattori,
The fixed point of a generalization of the renormalization group maps for
selfavoiding paths on gaskets,
Journal of Statistical Physics, 127 (2007) 609627.
160KB pdf file

T. Hattori, T. Tsuda,
Renormalization group analysis of the selfavoiding paths
on the ddimensional Sierpinski gaskets,
Journal of Statistical Physics 109 (2002) 3966.
350KB pdf file

K. Hattori, T. Hattori, S. Kusuoka,
Selfavoiding paths on the three dimensional Sierpinski gasket,
Publications of RIMS 29 (1993) 455509.
400KB pdf file

T. Hattori, S. Kusuoka,
The exponent for mean square displacement of selfavoiding
random walk on Sierpinski gasket,
Probability Theory and Related Fields 93 (1992) 273284.

K. Hattori, T. Hattori,
Selfavoiding process on the Sierpinski gasket,
Probability Theory and Related Fields 88 (1991) 405428.

K. Hattori, T. Hattori, S. Kusuoka,
Selfavoiding paths on the preSierpinski gasket,
Probability Theory and Related Fields 84 (1990) 126.
Introductory articles and seminar presentations are in the Japanese pages.
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