Local Bézout Theorem

AbstractWe give an elementary proof of what we call the Local Bézout Theorem. Given a system of n polynomials in n indeterminates with coefficients in a Henselian local domain, (V,m,k), which residually defines an isolated point in kn of multiplicity r, we prove (under some additional hypothesis on V) that there are finitely many zeroes of the system above the residual zero (i.e., with coordinates in m), and the sum of their multiplicities is r. Our proof is based on techniques of computational algebra

Identifying asymmetric, multi-period Euler equations estimated by non-linear IV/GMM

In this article, the identification of instrumental variables and generalized method of moment (GMM) estimators with multi-period perceptions is discussed. The state space representation delivers a conventional first order condition that is solved for expectations when the Generalized Bézout Theorem holds. Here, it is shown that although weak instruments may be enough to identify the parameters of a linearized version of the Quasi-Reduced Form (Q-RF), their existence is not sufficient for the identification of the structural model. Necessary and sufficient conditions for local identification of the Quasi-Structural Form (Q-SF) derive from the product of the data moments and the Jacobian. Satisfaction of the moment condition alone is only necessary for local and global identification of the Q-SF parameters. While the conditions necessary and sufficient for local identification of the Q-SF parameters are only necessary to identify the expectational model that satisfies the regular solution. If the conditions required for the decomposition associated with the Generalized Bézout Theorem are not satisfied, then limited information estimates of the Q-SF are not consistent with the full solution. The Structural Form (SF) is not identified in the fundamental sense that the Q-SF parameters are not based on a forward looking expectational model. This suggests that expectations are derived from a forward looking model or survey data used to replace estimated expectations

Existence and Uniqueness of Perturbation Solutions to DSGE Models

We prove that standard regularity and saddle stability assumptions for linear approximations are sufficient to guarantee the existence of a unique solution for all undetermined coefficients of nonlinear perturbations of arbitrary order to discrete time DSGE models. We derive the perturbation using a matrix calculus that preserves linear algebraic structures to arbitrary orders of derivatives, enabling the direct application of theorems from matrix analysis to prove our main result. As a consequence, we provide insight into several invertibility assumptions from linear solution methods, prove that the local solution is independent of terms first order in the perturbation parameter, and relax the assumptions needed for the local existence theorem of perturbation solutions.Perturbation, matrix calculus, DSGE, solution methods, Bézout theorem; Sylvester equations

Improved Complexity Bounds for Counting Points on Hyperelliptic Curves

We present a probabilistic Las Vegas algorithm for computing the local zeta function of a hyperelliptic curve of genus $g$ defined over $\mathbb{F}_q$. It is based on the approaches by Schoof and Pila combined with a modeling of the $\ell$-torsion by structured polynomial systems. Our main result improves on previously known complexity bounds by showing that there exists a constant $c>0$ such that, for any fixed $g$, this algorithm has expected time and space complexity $O((\log q)^{cg})$ as $q$ grows and the characteristic is large enough.Comment: To appear in Foundations of Computational Mathematic