David G. Glynn

The permanent of a square matrix

We investigate the permanent of a square matrix over a field and calculate it using ways different from Rysers formula or the standard definition. One formula is related to symmetric tensors and has the same efficiency O(2^mm) as Rysers method. Another algebraic method in the prime characteristic case uses partial differentiation.

The non-classical 10-arc of PG(4,9)

Abstract It is shown that PG(4,9) contains a non-classical 10-arc. It is the first example of a ( q + 1)-arc of PG(n, q), ( q ood, 2⩽ n ⩽ q −2), which is not a normal rational curve. Various properties of the arc are also derived.

On a set of lines of PG(3, q) corresponding to a maximal cap contained in the Klein quadric of PG (5, q)

The problem is considered of constructing a maximal set of lines, with no three in a pencil, in the finite projective geometry PG(3, q) of three dimensions over GF(q). (A pencil is the set of q+1 lines in a plane and passing through a point.) It is found that an orbit of lines of a Singer cycle of PG(3, q) gives a set of size q3 + q2 + q + 1 which is definitely maximal in the case of q odd. A (q3 + q2 + q + 1)-cap contained in the hyperbolic (or ‘Klein’) quadric of PG(5, q) also comes from the construction. (A k-cap is a set of k points with no three in a line.) This is generalized to give direct constructions of caps in quadrics in PG(5, q). For q odd and greater than 3 these appear to be the largest caps known in PG(5, q). In particular it is shown how to construct directly a large cap contained in the Klein quadric, given an ovoid skew to an elliptic quadric of PG(3, q). Sometimes the cap is also contained in an elliptic quadric of PG(5, q) and this leads to a set of q3 + q2 + q + 1 lines of PG(3,q2) contained in the non-singular Hermitian surface such that no three lines pass through a point. These constructions can often be applied to real and complex spaces.

The Conjectures of Alon-Tarsi and Rota in Dimension Prime Minus One

A formula for Glynns hyperdeterminant

A condition for the existence of ovals in PG(2, q), q even

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Rings of geometries II

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On the classification of geometric codes by polynomial functions

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The modular counterparts of Cayley's hyperdeterminants

prime) of a square matrix shows that the number of ways to decompose any integral doubly stochastic matrix with row and column sums

On the Anti-Pappian 103 and its Construction

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Ovoids and monomial ovals

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