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We investigate the probability that a random quadratic form in${\mathbb{Z}}[x_1,...,x_n]$ has a totally isotropic subspace of a givendimension. We show that this global probability is a product of localprobabilities. Our main result computes these local probabilities for quadraticforms over the $p$-adics. The formulae we obtain are rational functions in $p$invariant upon substituting $p \mapsto 1/p$.

The density of integral quadratic forms having a $k$-dimensional totally isotropic subspace