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Given a closed positive current$T$on a bounded Runge open subset Ω of$C$^{$n$}, we study sufficient conditions for the existence of a global extension of$T$to$C$^{$n$}. When$T$has a sufficiently low density, we show that the extension is possible and that there is no propagation of singularities, i.e.$T$may be extended by a closed positive$C$^{∞}-form outside $$\bar \Omega $$ . Conversely, using recent results of$H.$Skoda and$H.$El Mir, we give examples of non extendable currents showing that the above sufficient conditions are optimal in bidegree (1, 1).

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For any proper polynomial map$f: C$^{$k$}→$C$^{$k$}define the function α as $$\alpha (z): = \mathop {\lim \sup }\limits_{n \to \infty } \frac{{\log ^ + \log ^ + \left| {f^n (z)} \right|}}{n},where log^ + : = \max \{ \log , 0\} .$$ Let$f=(P$_{1},...,$P$_{$k$}) be a proper polynomial map. We define a notion of$s$-regularity using the extension of$f$to P^{k}. When$f$is (maximally) regular we show that the function α is lower semicontinuous and takes only finitely many values: 0 and$d$_{1},...,$d$_{$k$}, where$d$_{$i$}:=deg$P$_{$i$}. We then describe dynamically the sets {α≤$d$_{$i$}}. We give a concrete description of regular maps. If$d$_{$i$}>1, this allows us to construct the equilibrium measure μ associated with$f$as a generalized intersection of positive currents. We then give an estimate of the Hausdorff dimension of μ. We extend the approach to the larger class of (π,$s$)-regular maps. This gives an understanding of the largest values of α. The results can be applied to construct dynamically interesting measures for automorphisms.

Recent developments in multivariate polynomial solving algorithms have made algebraic cryptanalysis a plausible threat to many cryptosystems. However, theoretical complexity estimates have shown this kind of attack unfeasible for most realistic applications. In this paper we present a strategy for computing Gröbner basis that challenges those complexity estimates. It uses a flexible partial enlargement technique together with reduced row echelon forms to generate lower degree elements–mutants. This new strategy surpasses old boundaries and obligates us to think of new paradigms for estimating complexity of Gröbner basis computation. The new proposed algorithm computed a Gröbner basis of a degree 2 random system with 32 variables and 32 equations using 30 GB which was never done before by any known Gröbner bases solver.