The case when all the terms of W has y-degree 2 or less.

Proposition (19990221).

Let f, g, h be polynomials in x with non-negative coefficients, and with each term of order no less than 3, 2, 1, respectively. Assume, in particular, that g is not equivalent to 0. Put W(x,y) = f(x) + g(x) y + h(x) y². Then there is at most 1 critical point for w(x,y) = W(x,y)-(x²+y²)/2 (i.e., fixed point of grad W).

Remark.

The conditions for f, g, h in the Proposition are obviously satisfied under the assumption 1 of Problem 1 and the assumption that W has terms of y-degree 1, hence this Proposition solves the uniqueness problem completely if all the terms of W has y-degree 2 or less.

Proof.

A fixed point of grad W satisfies
x=W_x(x,y)=f'(x)+g'(x) y+h''(x) y²
and
y=W_y(x,y)=g(x)+2 h(x) y.

The latter equation implies that a fixed point exists (in the first quadrant) only for those x satisfying 2 h(x) < 1 . Then y(x):= g(x)/(1-2 h(x)) is increasing, and consequently, W_x(x,y(x)) =f'(x)+g'(x) y(x)+h''(x) y(x)² also is increasing. The assumptions also imply that W_x(x,y(x)) =O(x³) as x approaches 0 . Therefore x=W_x(x,y(x)) has at most 1 solution in x> 0 . This implies the Proposition.