4. The first computational phase: explicit formulas
The purpose of this phase was to determine a basic form for the test function, and especially to decide which choices of the background \(S\) and detector \(P\) in (2.5) were most effective. We focused first on \(\ell=2\), asking which functions \(G(\nu_1,\nu_2)\) could survive numerical screening.
4.1 The first numerical candidates
Put \(S=1-\nu_1^2-\nu_2^2\). The first ten tested candidates are collected below. Most belong to the \(\ell=2\) search; the final two are the immediate higher-rank product variants tested in the same phase.
| Candidate \(G=g^2\) | Numerical screening |
|---|---|
| \(G=1-\nu_1^2-\nu_2^2+k(\cos\vartheta-\nu_1)^2\) | Passed |
| \(G=S+k\nu_1^2\nu_2^2\) | Failed |
| \(G=S^2+k(\nu_1-\nu_2)^2(\nu_1+\nu_2-1)^2\) | Failed |
| \(G=\bigl[S^2+k(\nu_1-\nu_2)^2(\nu_1+\nu_2-1)^2\bigr]^{1/2}\) | Failed |
| \(G=\left[\prod_{\pm}\left(1-\frac{\sqrt{3}}{2}\nu_1\pm\frac{1}{2}\nu_2\right)(1+\nu_2)\right]^{2/3}\) | Passed |
| \(G=\left[S^{3/2}+k\prod_{\pm}\left(1-\frac{\sqrt{3}}{2}\nu_1\pm\frac{1}{2}\nu_2\right)(1+\nu_2)\right]^{2/3}\) | Failed |
| \(G=S+k(1-\nu_1^2)(1-\nu_2^2)\) | Failed |
| \(G=\bigl[S^2+k(1-\nu_1^2)(1-\nu_2^2)\bigr]^{1/2}\) | Failed |
| \(G=\bigl[(1-\nu_1)(1-\nu_2)(1-\nu_3)\bigr]^{2/3}\) | Passed |
| \(G=\bigl[(1-\nu_1^2)(1-\nu_2^2)(1-\nu_3^2)(1-\nu_4^2)\bigr]^{1/4}\) | Failed |
These numerical screenings ruled out many natural alternatives and led us to focus on candidates of the form \(G=S+kP\), where \(S=1-\nu_1^2-\nu_2^2\) and \(P\) vanishes at the prescribed directions. In particular, several simple choices, including \(P=\nu_1^2\nu_2^2\) and \(P=(1-\nu_1^2)(1-\nu_2^2)\), failed the numerical screening. This made clear that \(P\) had to be designed more carefully in order to obtain a viable candidate.
4.2 Searching over the detector \(P\)
The next round of experiments focus on \(\ell=3, 4\). Thus the background was
We varied the structure of \(P\), testing products of affine pole factors, fractional powers, repeated factors, and tilted factors.
| Candidate \(P\) | Numerical screening |
|---|---|
| Rank-three tests: \(S=1-\nu_1^2-\nu_2^2-\nu_3^2,\quad t\in [0,2\pi]\) | |
| \(P=(1-\nu_1)(1-\nu_2)\) | Passed |
| \(P=\bigl[(1-\nu_1)(1-\nu_2)(1-\nu_1/\sqrt{2}-\nu_3/\sqrt{2})\bigr]^{2/3}\) | Failed* |
| \(P=\bigl[(1-\nu_1)(1-\nu_2)(1-\nu_1\cos t-\nu_3\sin t)\bigr]^{2/3}\) | Failed* |
| \(P=\bigl[(1-\nu_1)^2(1-\nu_2)\bigr]^{2/3}\) | Failed* |
| \(P=\bigl[(1-\nu_1)(1-\nu_2)(1-\nu_3)(1-\nu_1\cos t-\nu_3\sin t)\bigr]^{1/2}\) | Failed* |
| \(P=\bigl[(1-\nu_1)^2(1-\nu_2)(1-\nu_3)\bigr]^{1/2}\) | Failed* |
| \(P=\bigl[(1-\nu_1)^2(1-\nu_2)(1-\nu_2\cos t-\nu_3\sin t)\bigr]^{1/2}\) | Failed* |
| \(P=\left[\prod_{i=1}^2(1-\nu_i)(1-\nu_i\cos t-\nu_3\sin t)\right]^{1/2}\) | Failed* |
| \(P=\bigl[(1-\nu_1^2)(1-\nu_2^2)(1-\nu_3^2)\bigr]^{1/3}\) | Failed* |
| Additional \(n=6\) tests: \(S=1-\nu_1^2-\nu_2^2-\nu_3^2-\nu_4^2\) | |
| \(P=\frac{1}{4}\bigl[(1-\nu_1)(1-\nu_2)(1-\nu_3)(1+\nu_3)\bigr]^2\) | Failed |
| \(P=(1-\nu_1)(1-\nu_2)(1-\nu_3)\) | Failed |
| \(P=(1-\nu_1)(1-\nu_2)\) | Failed |
Interpreting the starred failures.
Every formula marked “Failed\(^{*}\)” failed for the same localized reason: negative values were detected only in thin neighborhoods of one or more prescribed directions. On the sampled regions bounded away from the zeros, the numerical margin remained positive. As \(k\) decreased, the bad neighborhoods became smaller and were therefore easy to miss at fixed numerical resolution. An unstarred “Failed” means that negative values were already visible in the retained general scan, rather than only in a zero neighborhood.
Thus the experiments were almost entirely negative, but the location of the failure was encouraging: away from the zeros, the starred candidates retained a positive margin. This suggested that the global product structure might still be useful and that only the local vanishing model had to be repaired. Even a sheeting theorem for three arbitrary prescribed directions would already have been a worthwhile result, but at this stage no choice of \(P\) passed all the relevant tests.
A second pattern identified the exponent itself. Every starred candidate shared the balanced power structure
with repeated or tilted factors allowed.
Beyond the formulas listed in the table, we numerically tested many products with other exponents and several non-power modifications. These additional candidates developed negative values away from the zero set. By contrast, the observed failures of the balanced \(2/m\)-power products were confined to shrinking neighborhoods of the zeros.
The data now pointed both to a candidate exponent and to a rank restriction. While the balanced products behaved consistently in ranks two and three, the rank-four experiments were markedly less stable: new negative directions appeared away from the zeros, and neither decreasing \(k\) nor adding more factors removed them. We therefore restricted the next stage of the search to \(\ell\leq3\) and replaced the open-ended search for new formulas by an analysis of the necessary conditions on \(P\). In particular, we examined the regime \(k\downarrow0\) to determine whether the exponent \(2/m\) was indeed sufficient on every compact set away from the zeros, to understand why the bad neighborhoods shrank, and to isolate the additional local condition that would eventually be needed at a zero. This change of viewpoint led directly to the next section.
Gaoming Wang
Assistant Professor
My research interests include Geometric Analysis and Partial Differential Equations.