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# Probabilistic Diophantine geometry

## Efthymios Sofos

Let $$F$$ be a smooth homogeneous polynomial with integer coefficients.

In the first part we will see that the prime factorisation of the coordinates of the integer points on $$F=0$$ can recognise a part of the geometry of $$F=0$$. Namely, we shall model the factorisation by a multivariate Gaussian distribution and we will see that the covariance matrix of the Gaussian distribution (which is a statistical invariant) keeps track of certain geometric data related to the dimensions of hyperplane sections of $$F$$.

In the second part we will study the real numbers given by $$(\log\log p_i)/(\log \log H)$$, where $$p_1\lt p_2\lt \dots\lt p_k$$ are the primes for which $$F=0$$ has no $$p$$-adic zeros and $$H$$ is the size of the coefficients of $$F$$. We will see that these numbers are modelled by $$k$$ uniformly distributed random points in the interval $$[0,1]$$, as long as $$F$$ ranges in an infinite family. This leads to several connections between local solubility for varieties and normal distribution, Brownian motion and Poisson point process.