18 research outputs found
Rigid G2-Representations and motives of Type G2
We prove an effective Hilbert Irreducibility result for residual realizations
of a family of motives with motivic Galois group G2
Topology and Factorization of Polynomials
For any polynomial , we describe a
-vector space of solutions of a linear system of equations
coming from some algebraic partial differential equations such that the
dimension of is the number of irreducible factors of . Moreover, the
knowledge of gives a complete factorization of the polynomial by
taking gcd's. This generalizes previous results by Ruppert and Gao in the case
.Comment: Accepted in Mathematica Scandinavica. 8 page
Computational Arithmetic Geometry I: Sentences Nearly in the Polynomial Hierarchy
We consider the average-case complexity of some otherwise undecidable or open
Diophantine problems. More precisely, consider the following: (I) Given a
polynomial f in Z[v,x,y], decide the sentence \exists v \forall x \exists y
f(v,x,y)=0, with all three quantifiers ranging over N (or Z). (II) Given
polynomials f_1,...,f_m in Z[x_1,...,x_n] with m>=n, decide if there is a
rational solution to f_1=...=f_m=0. We show that, for almost all inputs,
problem (I) can be done within coNP. The decidability of problem (I), over N
and Z, was previously unknown. We also show that the Generalized Riemann
Hypothesis (GRH) implies that, for almost all inputs, problem (II) can be done
via within the complexity class PP^{NP^NP}, i.e., within the third level of the
polynomial hierarchy. The decidability of problem (II), even in the case m=n=2,
remains open in general.
Along the way, we prove results relating polynomial system solving over C, Q,
and Z/pZ. We also prove a result on Galois groups associated to sparse
polynomial systems which may be of independent interest. A practical
observation is that the aforementioned Diophantine problems should perhaps be
avoided in the construction of crypto-systems.Comment: Slight revision of final journal version of an extended abstract
which appeared in STOC 1999. This version includes significant corrections
and improvements to various asymptotic bounds. Needs cjour.cls to compil
Improvements on dimension growth results and effective Hilbert's irreducibility theorem
We sharpen and generalize the dimension growth bounds for the number of
points of bounded height lying on an irreducible algebraic variety of degree
, over any global field. In particular, we focus on the the affine
hypersurface situation by relaxing the condition on the top degree homogeneous
part of the polynomial describing the affine hypersurface. Our work sharpens
the dependence on the degree in the bounds, compared to~\cite{CCDN-dgc}. We
also formulate a conjecture about plane curves which gives a conjectural
approach to the uniform degree case (the only case which remains open). For
induction on dimension, we develop a higher dimensional effective version of
Hilbert's irreducibility theorem.Comment: 35 page
Solving Degenerate Sparse Polynomial Systems Faster
Consider a system F of n polynomial equations in n unknowns, over an
algebraically closed field of arbitrary characteristic. We present a fast
method to find a point in every irreducible component of the zero set Z of F.
Our techniques allow us to sharpen and lower prior complexity bounds for this
problem by fully taking into account the monomial term structure. As a
corollary of our development we also obtain new explicit formulae for the exact
number of isolated roots of F and the intersection multiplicity of the
positive-dimensional part of Z. Finally, we present a combinatorial
construction of non-degenerate polynomial systems, with specified monomial term
structure and maximally many isolated roots, which may be of independent
interest.Comment: This is the final journal version of math.AG/9702222 (``Toric
Generalized Characteristic Polynomials''). This final version is a major
revision with several new theorems, examples, and references. The prior
results are also significantly improve
A note on Gao’s algorithm for polynomial factorization
AbstractShuhong Gao (2003) [6] has proposed an efficient algorithm to factor a bivariate polynomial f over a field F. This algorithm is based on a simple partial differential equation and depends on a crucial fact: the dimension of the polynomial solution space G associated with this differential equation is equal to the number r of absolutely irreducible factors of f. However, this holds only when the characteristic of F is either zero or sufficiently large in terms of the degree of f. In this paper we characterize a vector subspace of G for which the dimension is r, regardless of the characteristic of F, and the properties of Gao’s construction hold. Moreover, we identify a second vector subspace of G that leads to an analogous theory for the rational factorization of f