Suppose I told you that by further restricting the variables to lie on the unit circle, then the solutions would be isolated. Would that help? These solutions would fill a gap in the classification of order 7 complex Hadamard matrices. A complex Hadamard matrix is a unitary matrix with equal modulus entries. They are usually rescaled so the entries have modulus 1. A complex Hadamard matrix that happens to be real is the more familiar Hadamard matrix, which can exist only when the order is 1, 2, or a multiple of 4. All the known complex Hadamard matrices of order less than 11 have the property that their entries are constructed by taking products of integer powers of a small set of generators of modulus 1. Something strange happens at order 11, where it appears that there are thousands of isolated complex Hadamard matrices without this structure. I've computed some of these to over 100 digits and looked for integer relations among the phase angles using LLL and found none -- an approach to which all the lower order matrices easily yield. Does this elevate your respect for the number 11? Veit On Aug 24, 2011, at 8:19 AM, Victor Miller wrote:
On Wed, Aug 24, 2011 at 1:56 AM, Bill Gosper <billgosper@gmail.com> wrote:
Victor Miller> Veit, I gave this to SAGE (which actually uses the Groeber Basis stuff
in SINGULAR) and it fairly quickly calculated a Groebner Basis, and showed that the dimension of the ideal is 3.
I.e., triply underdetermined?
Yep, That's what it means.
That scotches the plan to PSLQ the exact algebraics from 1000 digit approximations. --rwg
Right now I'm waiting for it to produce a primary decomposition which should shed some light on the matter.
Victor
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