We formalize the Berlekamp-Zassenhaus algorithm for factoring square-free integer polynomials in Isabelle/HOL. We further adapt an existing formalization of Yun’s square-free factorization algorithm to integer polynomials, and thus provide an efficient and certified factorization algorithm for arbitrary univariate polynomials. The algorithm first performs a factorization in the prime field GF(p...

Chen's biharmonic conjecture is well-known and stays open: The only biharmonic submanifolds of Euclidean spaces are the minimal ones. In this paper, we consider an advanced version of the conjecture, replacing $Delta$ by its extension, $L_1$-operator ($L_1$-conjecture). The $L_1$-conjecture states that any $L_1$-biharmonic Euclidean hypersurface is 1-minimal. We prove that the $L_1$-conje...

As reported by A.I. Kostrikin in his paper [21], the very first algebra which distinguished modular (i.e. over fields of positive characteristic p) Lie algebra theory from the classical one was the Witt algebra W (1 : k), where k is a power of p. This algebra is a generalization due to H. Zassenhaus [31] in the thirties of an analogous structure defined by E. Witt over the integers. This algebr...

Although a polynomial time algorithm exists, the most commonly used algorithm for factoring a univariate polynomial f with integer coeecients is the Berlekamp-Zassenhaus algorithm which has a complexity that depends exponentially on n where n is the number of modular factors of f. This exponential time complexity is due to a combinatorial problem; the problem of choosing the right subset of the...