By Damascelli L., Gladiali F.

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Xn ) ≥ 0 for all (x1 , x2 , . . , xn ) ∈ Rn , can one find a set of s real polynomials Qk (x1 , x2 , . . , xn ), 1 ≤ k ≤ s, such that P (x1 , x2 , . . , xn ) = Q21 + Q22 + · · · + Q2s ? This problem turns out to be wonderfully rich. It leads to work that is deeper and more wide ranging than our earlier problems, and, even now, it continues to inspire new research. A Definitive Answer — In a Special Case As usual, one does well to look for motivation by examining some simple cases. Here the first case that is not completely trivial occurs when n = 1 and the polynomial P (x) is simply a quadratic ax2 + bx + c with a = 0.

Almost without help, it pulls itself up by the bootstraps to a new result which covers cases that were left untouched by the original. Under normal circumstances, this generalization might seem to be too easy to qualify as a challenge problem, but the final result is so important the problem easily clears the hurdle. 2 (The AM-GM Inequality with Rational Weights) Suppose that p1 , p2 , . . , pn are nonnegative rational numbers that sum to one, and show that for any nonnegative real numbers a1 , a2 , .

Minkowski’s conjecture is now on the ropes; we just need to land a knock-out punch. When we look back at our candidate P (x, y), we see the striking feature that all of its coefficients are nonnegative except for the coefficient of x2 y 2 which is equal to −3. This observation suggests that we should see what one can say about the possible values of the coefficient of x2 y 2 in the sum Q21 (x, y) + Q22 (x, y) + · · · + Q2s (x, y). Here we have some genuine luck. 9) of the terms Qk (x, y), 1 ≤ k ≤ s, we can easily check that the coefficient of x2 y 2 in the polynomial Q21 (x, y) + Q22 (x, y) + · · · + Q2s (x, y) is just b21 + b22 + · · · + b2s .

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Some new entire solutions of semilinear elliptic equations on Rn by Damascelli L., Gladiali F.


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