Browsing by Author "Hughes, Kenneth R"

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    Aspects of higher degree forms with symmetries
    (1996) Omar, Mohammed Rafiq; Hughes, Kenneth R
    In Chapter One we develop a basis for studying higher degree alternating forms. The concepts and results we present are mostly obvious analogues of Harrison's treatment of higher degree symmetric forms. We explain antisymmetrization; discuss the derivative of an alternating form and its corresponding anticommutative polynomial; define alternating spaces and their direct sum; establish decomposition and cancellation results for alternating spaces; and construct a Witt-Grothendieck group of alternating spaces. In Chapter Two we discuss hyperbolic alternating space. We compute the centre, algebraic isometry group and its corresponding Lie algebra, and prove a descent result. There are important parallels with Keet's results for hyperbolic symmetric spaces, as well as significant differences, especially in the methods we employ. In Chapter Three we develop a framework for the study of two aspects of forms of general Young symmetry type: their hyperbolics, and a generalization of the Weil-Siegel duality between symmetric and alternating bilinear forms. We introduce notions like nondegeneracy, derivative of a form, and derivative and integral symmetry types, and are then able to construct a hyperbolic space which is cofinal for spaces equipped with a form of the same symmetry type, and show that symmetry types are Siegel duals in our generalized sense if they have the same derivative symmetry type. In Chapter Four we present a few results and observations concerning nondegeneracytype conditions on symmetric forms. These include: an extension of Harrison's proof that nonsingularity implies nonzero Hessian to forms of arbitrary degree; a discussion of s-nondegeneracy and s-regularity; and a relation between a strong nondegeneracy condition on forms of even degree and the catalecticant, a classical invariant.
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    On the theory of Krull rings and injective modules
    (1988) Prince, R N; Hughes, Kenneth R
    In the first chapter we give an outline of classical KRULL rings as in SAMUEL (1964), BOURBAKI (1965) and FOSSUM (1973). In the second chapter we introduce two notions important to our treatment of KRULL theory. The first is injective modules and.the second torsion theories. We then look at injective modules over Noetherian rings as in MATLIS [1958] and then over KRULL rings as in BECK [1971]. We show that for a KRULL ring there is a torsion theory (N,M) where N is the pseudo-zero modules and M the set of N-torsion-free (BECK calls these co-divisorial) modules. From LAMBEK [1971] there is a full abelian sub category C, namely the category of N-torsion-free, N-divisible modules, with exact reflector. We show in C (I) every direct sum of injective modules is injective and (II) C has global dimension at most one. It is these two properties that we exploit in the third chapter to give another characterization of KRULL rings. Then we generalize this to rings with zero-divisors and find that (i) R has to be reduced (ii) the ring is KRULL if and only if it is a finite product of fields and KRULL domains (iii) the injective envelope of the ring is semi-simple artinian. We then generalize the ideas to rings of higher dimension.
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    Topics in the algebraic theory of higher degree forms
    (1991) Keet, Arnold Peter; Hughes, Kenneth R
    Let d≥2 be an integer and let F be a field. A form of degree d over F is a polynomial of homogeneous degree d with coefficients in F. In degree d=2 there is an extensive theory of quadratic forms. We consider forms of degree d>2. The following are among the new results we have proved: 1. A nonsingular form over a field of characteristic zero has nonzero Hessian. This was proved by Harrison in degree d=3. We use some basic algebraic geometry and rational differential forms to give a proof valid in all degrees d≥2. 2. The formal differences of split forms constitute an ideal in the Grothendieck ring of higher degree forms. This generalises a well-known result for quadratic forms to higher degree. 3. In the monoid of equivalence classes of nondegenerate forms of degree d≥3, with the tensor product operation, the submonoid generated by the equivalence classes of the hyperbolic forms is free. This is a small step towards answering a question posed by Harrison. 4. Let the base field have characteristic zero. In every odd degree d≥3 there are no nontrivial families of additive invariants of the forms of degree d. In every even degree d there is a nontrivial family of additive invariants of the forms of degree d. The most familiar example is the family of discriminants of quadratic forms. Our proof involves the symbolic method for representing invariants of the forms of degree d. 5. We give a new proof, in characteristic zero, that a nonsingular form of degree d≥3 has a zero Lie algebra. Our proof involves a certain Schur functor and invokes the basis theorem of Akin, Buchsbaum and Weyman. 6. Let F be a field of characteristic zero and let K:F be a field extension. Let f be a form of degree d≥3 in more than three indeterminates with coefficients in F. Then if f is equivalent over K to a hyperbolic form, f must already be equivalent to a hyperbolic form over F. Compare this with the degree 2 case where, for example, the forms Σ(xk²+yk²) are hyperbolic over C but not over R.