On pseudorandom numbers from multivariate polynomial systems
This paper deals with sequences of vectors \(\mathbf{u}_n=(u_{n,1,},\ldots,u_{n,m})\in\mathbb{F}_p^m\) generated by multivariate polynomial systems. The vectors \(\mathbf{u}_n\) are defined by a recurrence congruence modulo the prime \(p\), \[ u_{n+1,i}=f_i(u_{n,1},\ldots,u_{n,m}),\;n=0,1,\ldots, \] with initial values \(\mathbf{u}_0=(u_{0,1},\ldots,u_{0,m})\), where \(f_1(X_1,\ldots,X_m),\ldots,f_m (X_1,\ldots,X_m)\) are polynomials over \(\mathbb{F}_p\). The authors derive bounds on exponential sums along the orbits of such polynomial systems. These bounds hold for essentially arbitrary polynomial systems with sufficiently long orbits, and they can be applied, via the Erdős-Turán-Koksma inequality, to deriving discrepancy bounds for sequences of pseudorandom vectors based on the sequence of the \(\mathbf{u}_n\). The approach in this paper, in comparison to previous results, does not require control of the degree growth of the iterations of the polynomial system.
scientific article; zbMATH DE number 5798562
scientific article