Abstract:
Projective spaces over local ring $R=2R$ with principal maximal ideal $J,$$1+J\subseteq R^{*2}$ have been investigated. Quadratic forms and corresponding symmetric matrices $A$ and $B$ are projectively congruent if $kA = UBU^T$ for a matrix $U \in GL(n,R)$ and for some $k \in R^{*}.$ In the case of $k=1$ quadratic forms (corresponding symmetric matrices) are called congruent. The problem of enumerating congruent and projective congruent quadratic forms is based on the identification of the (unique) normal form of the corresponding symmetric matrices and is related to the theory of quadratic form schemes. Over the local ring $R$ on conditions $R^{*}/R^{*2}\!=\!\lbrace 1, -1, p,-p\rbrace$ and $D(1,1)\!=\!D(1,p)\!=\!\lbrace 1,p\rbrace,$ $D(1,-1)\!=\!D(1,-p)\!=\!\lbrace 1,-1,p,-p\rbrace$ (unique) normal form of congruent symmetric matrices over ring $R$ is detected. Quantities of congruent and projective congruent symmetric matrix classes is found when maximal ideal is nilpotent.
Keywords:projective spaces, local rings, projective congruence, projective equivalence.
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