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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 <= x2 } 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:

  1. W is a non-empty sum of terms each of total degree 3 or higher and with positive coefficients.
  2. grad W(D) is a subset of D - {(x,y) in D | y = x2 }. Here, grad W=(Wx,Wy): R+2 -> R+2 is the gradient map of W. (Wx is a partial derivative of W by x and Wy is that by y.)
  3. W(x,y) contains a term linear in y (a term of a form xny for some n>= 2). In other words, Y(x,0)=Wy(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)-(x2+y2)/2 defined by grad w(x,y)=0, uniquely exists in D.

Remarks.

  1. Critical point of the polynomial w=W-(x2+y2)/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=x2 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.
  2. 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.
  3. 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) 1-26) 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

  1. converge to the origin,
  2. diverge to the infinity,
  3. 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).

Random walk and renormalization group - An introduction to mathematical physics, by T.Hattori 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.)

  1. K. Hattori, T. Hattori, Sh. Kusuoka, Self-avoiding paths on the three dimensional Sierpinski gasket, Publications of RIMS 29 (1993) 455-509
  2. T. Hattori, T. Tsuda, Renormalization group analysis of the self-avoiding paths on the d-dimensional Sierpinski gaskets, Journal of Statistical Physics 109 (2002) 39--66.

(2) Approach from inverse function theorem.

By adding rather weak assumptions stating one-dimensional-like 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:

  1. R(x,z)=X2(x,x2z)-Y(x,x2z) is a polynomial in z, 1-z, and x, with non-negative coefficients.
  2. It holds that R(x,x2z)/Y(x,x2z) = 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.

  1. I called the first two conditions one-dimensional-like, in the sense that under these conditions, Y is close to X2 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=x2, so that the original picture that y is of higher order compared to x becomes clearer.
  2. 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 y-degree 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 y-degree 1 and y-degree 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 self-avoiding 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, non-zero, coefficient) a term x3. 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.

  1. W(x,y) contains a term x3 (i.e., its coefficient is poositive non-zero).
  2. Each term in W(x,y) has a total (x plus y) degree 6 or less.
  3. The terms with positive powers in y has total degrees 5 or more (i.e., 5 or 6).
  4. W contains no term of the forms xy4 or x2y3.
Then there uniquely exists a critical point in D of the polynomial w(x,y)= W(x,y)-(x2+y2)/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.

  1. W(x,y) = a x3 + b x4 + f5 x5 + f6 x6 + (3 a x2)2 y + g5 x5 y + h3 x3 y2 + h4 x4 y2 + n3 x3 y3 + a24 x2 y4 + a05 y5 + a15 x y5 + a06 y6, where the 12 coefficients a, b, f5, f6, g5, h3, h4, n3, a24, a05, a15, a06 are non-negative, and a>0.
  2. Rn, n=5,6,7,8,9,19, defined by the following, are non-negative.

W=W3 and W=W4 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

What is non-trivial?

Note that, restricting the possible terms of W to 12 types as in Proposition (20060819), the non-trivial 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=Wu just introduced,

  1. 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(u1/8), O(u-1/4)),
  2. 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.

generalization of RG for rSAP on dSG; global aspects of uniqueness of fixed point 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 non-trivial phenomena.

How come the conditions?

The uniqueness results were long known to hold for the examples W=W3 and W=W4, for which grad W are essentially the renormalization group for the restricted self-avoiding 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 W3 and W4 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 self-avoiding paths) in the background problems for W3 and W4.

Though the proof of Theorem (20060819) which we have does not use any intuition from the properties of self-avoiding paths, by yet-to-be-understood 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 self-avoiding 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 non-negative and that their combinations Rn are non-negative, are, in the examples W3 and W4, derived from the non-negativity of path numbers (which convey to non-negativity of probability measures) and self-avoiding 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.

  1. 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.
  2. Extensions to W with 3 or more variables are of course of interest. In fact, the background problem of the restricted self-avoiding paths on the d dimensional gakset corresponds to W with [(d+1)/2] variables.
  3. Existence of fixed points leads to continuum limit construction of continuous time self-similar processes with non-trivial fine structures (e.g., Haussdorff dimension greater than 1). Meanwhile, to prove asymptotic properties (e.g., displacement exponents and laws of iterated logarithms) for self-avoiding 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 Introductory articles and seminar presentations are in the Japanese pages.
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