Schur's transforms of a polynomial are used to count its roots in the unit disk. These are genera... more Schur's transforms of a polynomial are used to count its roots in the unit disk. These are generalized them by introducing the sequence of symmetric sub-resultants of two polynomials. Although they do have a determinantal definition, we show that they satisfy a structure theorem which allows us to compute them with a type of Euclidean division. As a consequence, a fast algorithm based on a dichotomic process and FFT is designed. We prove also that these symmetric sub-resultants have a deep link with Toeplitz matrices. Finally, we propose a new algorithm of inversion for such matrices. It has the same cost as those already known, however it is fraction-free and consequently well adapted to computer algebra.
In this paper we describe a fast algorithm for counting the number of roots of complex polynomial... more In this paper we describe a fast algorithm for counting the number of roots of complex polynomials in the open unit disk, classically called the Schur Cohn problem. For degree d polynomials our bound is of order d log 2 d in terms of field operations and comparisons, but our approach nicely allow us to integrate the bit size as a further complexity parameter as well. For bit size _ of input polynomials of Z[X] our running time is of order d 2 _ } log(d_) } log log(d_) } log(d) in the multitape Turing machine model, thus improving all previously proposed approaches.
Schur's transforms of a polynomial are used to count its roots in the unit disk. These are genera... more Schur's transforms of a polynomial are used to count its roots in the unit disk. These are generalized them by introducing the sequence of symmetric sub-resultants of two polynomials. Although they do have a determinantal definition, we show that they satisfy a structure theorem which allows us to compute them with a type of Euclidean division. As a consequence, a fast algorithm based on a dichotomic process and FFT is designed. We prove also that these symmetric sub-resultants have a deep link with Toeplitz matrices. Finally, we propose a new algorithm of inversion for such matrices. It has the same cost as those already known, however it is fraction-free and consequently well adapted to computer algebra.
In this paper we describe a fast algorithm for counting the number of roots of complex polynomial... more In this paper we describe a fast algorithm for counting the number of roots of complex polynomials in the open unit disk, classically called the Schur Cohn problem. For degree d polynomials our bound is of order d log 2 d in terms of field operations and comparisons, but our approach nicely allow us to integrate the bit size as a further complexity parameter as well. For bit size _ of input polynomials of Z[X] our running time is of order d 2 _ } log(d_) } log log(d_) } log(d) in the multitape Turing machine model, thus improving all previously proposed approaches.
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Papers by Cyril Brunie