Discovering the Roots: Uniform Closure Results for Algebraic Classes Under Factoring

Newton iteration is an almost 350-year-old recursive formula that approximates a simple root of polynomial quite rapidly. We generalize it to matrix recurrence (allRootsNI) all roots simultaneously. In this form, the process yields better circuit complexity in case when number r small but multiplicities are exponentially large. Our method sets up linear system unknowns and iteratively builds as formal power series. For algebraic \( f(x_1,\ldots ,x_n) \) size s , we prove each factor has at most degree squarefree part f . Consequently, if f_1 2^{\Omega (n)} -hard polynomial, then any nonzero multiple \prod _{i} f_i^{e_i} equally hard for arbitrary positive e_i ’s, assuming \sum _i\deg (f_i) 2^{O(n)} It old open question whether class poly( n ) formulas (respectively, branching programs) closed under factoring. show given n^{O(1)} program) n^{O(\log n)} can find similar-size randomized n^{\log n} time. determinant requires n^{\Omega (\log formula, same be said about its multiples. our proofs, exploit following property multivariate factorization. Under random transformation \tau f(\tau \overline{x}) completely factors via series roots. Moreover, factorization adapts well analysis. Therefore, with help strong mathematical characterizations ‘allRootsNI’ technique, make significant progress towards problems; supplementing vast body classical results concepts (e.g., [ 17 51 54 111 ]).