Discovering the roots: Uniform closure results for algebraic classes under factoring

Newton iteration (NI) is an almost 350 years old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all the roots simultaneously. In this form, the process yields a better circuit complexity in the case when the number of roots $r$ is small but the multiplicities are exponentially large. Our method sets up a linear system in $r$ unknowns and iteratively builds the roots as formal power series. For an algebraic circuit $f(x_1,\ldots,x_n)$ of size $s$ we prove that each factor has size at most a polynomial in: $s$ and the degree of the squarefree part of $f$. Consequently, if $f_1$ is a $2^{\Omega(n)}$-hard polynomial then any nonzero multiple $\prod_{i} f_i^{e_i}$ is equally hard for arbitrary positive $e_i$'s, assuming that $\sum_i \text{deg}(f_i)$ is at most $2^{O(n)}$. It is an old open question whether the class of poly($n$)-sized formulas (resp. algebraic branching programs) is closed under factoring. We show that given a polynomial $f$ of degree $n^{O(1)}$ and formula (resp. ABP) size $n^{O(\log n)}$ we can find a similar size formula (resp. ABP) factor in randomized poly($n^{\log n}$)-time. Consequently, if determinant requires $n^{\Omega(\log n)}$ size formula, then the same can be said about any of its nonzero multiples. As part of our proofs, we identify a new property of multivariate polynomial factorization. We show that under a random linear transformation $\tau$, $f(\tau\overline{x})$ completely factors via power series roots. Moreover, the factorization adapts well to circuit complexity analysis. This with allRootsNI are the techniques that help us make progress towards the old open problems, supplementing the large body of classical results and concepts in algebraic circuit factorization (eg. Zassenhaus, J.NT 1969, Kaltofen, STOC 1985-7 \& Burgisser, FOCS 2001).Comment: 33 Pages, No figure

Stark-Heegner points on modular jacobians

We present a construction which lifts Darmon’s Stark–Heegner points from elliptic curves to certain modular Jacobians. Let N be a positive integer and let p be a prime not dividing N. Our essential idea is to replace the modular symbol attached to an elliptic curve E of conductor Np with the universal modular symbol for Γ0(Np). We then construct a certain torus T over Qp and lattice L ⊂ T, and prove that the quotient T/Lis isogenous to the maximal toric quotient J0(Np) p-new of the Jacobian of X0(Np).This theorem generalizes a conjecture of Mazur, Tate, and Teitelbaum on the p-adic periods of elliptic curves, which was proven by Greenberg and Stevens. As a by-product of our theorem, we obtain an efficient method of calculating the p-adic periods of J0(Np) p-new

A Serre weight conjecture for geometric Hilbert modular forms in characteristic p

Let p be a prime and F a totally real field in which p is unramified. We consider mod p Hilbert modular forms for F, defined as sections of automorphic line bundles on Hilbert modular varieties of level prime to p in characteristic p. For a mod p Hilbert modular Hecke eigenform of arbitrary weight (without parity hypotheses), we associate a two-dimensional representation of the absolute Galois group of F, and we give a conjectural description of the set of weights of all eigenforms from which it arises. This conjecture can be viewed as a "geometric" variant of the "algebraic" Serre weight conjecture of Buzzard-Diamond-Jarvis, in the spirit of Edixhoven's variant of Serre's original conjecture in the case F = Q. We develop techniques for studying the set of weights giving rise to a fixed Galois representation, and prove results in support of the conjecture, including cases of partial weight one.Comment: Revised introduction, updated references, 71 page

BEAL’S CONJECTURE

This paper provides algebraic mathematical proof to 6 unsolved problems in mathematics (number theory) and 3 others. 1)       The Beal’s conjecture (invalid forever) 2)       The Wells summation theorem 3)       The Fermats last theorem (376yrs- invalid forever) 4)       Unification engine of  all Power  summations 5)       The Goldbach conjecture (271yrs-valid forever) 6)       The proof of solitary -10-(valid forever ) 7)       NP VS P-problem,[ANS=NP≠P =[ NP- UNTIL- P ]= [NP-R-EI-T-P] 8)       The Riemann Hypothesis (invalid forever) 9)       Power summation pyramid-Carnox pyramid Keywords: conjecture, vector analysis, three dimension, two dimension, increment, complex, law of algebra, prime factor, domain, algorithm, input, integer, real number line, dual set, magnitude, resultant, compute, binary, hack, twelve, shrink, standard, probability, linear equation, intersect, finite, 2% logarithmic, surds, narrow range, 98%, one dimensional space, compare, determine, flow, generalization,  infinity, independent, error, constant (k), dependent, graph, discretely, unaffected, HSIV, 4-input synchronous tetra- set, S4ISC, frequency, per binary input (BPI), aeroplane, run way, air friction, air resistance, take off angle, plane crash, geometry, optimum, slope, partially collapsed, totally collapsed, supremacy, incoherence, airflight study, name, equation index or factor, honour, constant factor, economics, start discontinuity, man, robot, creation, simultaneously, cage theory, like energies, function solute, trivial, non-trivial, unplu