This work deals with the generalized eigenvalue problem for nonsquare matrix pencils A − λB such that matrices A,B ∈ MC (m×n) show a given structure. More precisely, we assume they result from removing the first row of some matrix G ∈ MC ((m+1) , n) in the case of A, and its last row in the case of B. This structured generalized eigenvalue problem can be found in signal processing methods and i...

Let S be a set. For an arithmetical map ψ : Z → S, if for some n ∈ Z = {1, 2, 3, · · · } we have ψ(x+ n) = ψ(x) for all x ∈ Z, then we say that ψ is periodic modulo n and n is a period of ψ. Let P(S) denote the set of all periodic maps ψ : Z → S. If m,n ∈ Z are periods of a map ψ ∈ P(S), then the greatest common divisor (m,n) is also a period of ψ, for we can write (m,n) in the form am + bn wit...

This paper describes the application of the Sylvester resultant matrix to image deblurring. In particular, an image is represented as a bivariate polynomial and it is shown that operations on polynomials, specifically greatest common divisor (GCD) computations and polynomial divisions, enable the point spread function to be calculated and an image to be deblurred. The GCD computations are perfo...

Earlier results expressing multivariate subresultants as ratios of two subdeterminants of the Macaulay matrix are extended to Jouanolou’s resultant matrices. These matrix constructions are generalizations of the classical Macaulay matrices and involve matrices of significantly smaller size. Equivalence of the various subresultant constructions is proved. The resulting subresultant method improv...

The computation of the Greatest Common Divisor (GCD) of many polynomials is a nongeneric problem. Techniques defining “approximate GCD” solutions have been defined, but the proper definition of the “approximate” GCD, and the way we can measure the strength of the approximation has remained open. This paper uses recent results on the representation of the GCD of many polynomials, in terms of fac...

Let a, b and n be positive integers and let S = {x1, x2, . . . , xn} be a set of distinct positive integers. The n × n matrix (Sf ) = (f ((xi, xj))), having f evaluated at the greatest common divisor (xi, xj) of xi and xj as its ij−entry, is called the GCD matrix associated with f on the set S. Similarly, the n × n matrix [Sf ] = (f ([xi, xj ])) is called the LCM matrix associated with f on S. ...

Let (X,∆) be a proper dlt pair and L a nef Cartier divisor such that aL − (KX +∆) is nef and log big on (X,∆) for some a ∈ Z>0. Then |mL| is base point free for every m ≫ 0. 0. Introduction The purpose of this paper is to prove the following theorem. This type of base point freeness was suggested by M. Reid in [Re, 10.4]. Theorem 0.1 (Base point free theorem of Reid-Fukuda type). Assume that (X...

We demonstrate that the answer to the question posed in the title is “yes” and “no”: “no” if the set of permissible operations is restricted to {+,−,×,/,mod,<}; “yes” if we are also allowed a gcd-oracle as a permissible operation. It has been shown (see [Sto76, MST91]) that no strongly polynomial algorithm exists for the problem of finding the greatest common divisor (gcd) of two arbitrary inte...

J. J. Sylvester has announced formulas expressing the subresultants (or the successive polynomial remainders for the Euclidean division) of two polynomials, in terms of some double sums over the roots of the two polynomials. We prove Sylvester formulas using the techniques of multivariate polynomials involving multi-Schur functions and divided differences. Introduction and statement of the main...