https://hal.inria.fr/hal-01064519/document

Asymptotic implicit function theorems

On the complexity of the F 5 Gröbner basis algorithm

We study the complexity of Grobner bases computation, in particular in the generic situation where the variables are in simultaneous Noether position with respect to the system. 
We give a bound on the number of polynomials of degree $d$ in a Grobner basis computed by Faugere's $F_5$ algorithm~(Fau02) in this generic case for the grevlex ordering (which is also a bound on the number of polynomials for a reduced Grobner basis, independently of the algorithm used). Next, we analyse more precisely the structure of the polynomials in the Grobner bases with signatures that $F_5$ computes and use it to bound the complexity of the algorithm. 
Our estimates show that the version of~$F_5$ we analyse, which uses only standard Gaussian elimination techniques, outperforms row reduction of the Macaulay matrix with the best known algorithms for moderate degrees, and even for degrees up to the thousands if Strassen's multiplication is used. The degree being fixed, the factor of improvement grows exponentially with the number of variables.

On the Complexity of the F5 Gr\"obner basis Algorithm

We give constructive proofs for two Borel-Ritt interpolation results, stating the existence of holomorphic functions on polysectors admitting an arbitrarily prescribed asymptotic expansion, whether in the sense of Gerard-Sibuya or in that of Majima. As a consequence, a new proof is obtained for the classical Borel's theorem on the existence of C 1 functions onR n with given derivatives at 0; in fact, our construction provides functions analytic in (R {0}) n .

Asymptotic implicit function theorems

Citations

On the complexity of the F 5 Gröbner basis algorithm

On the Complexity of the F5 Gr\"obner basis Algorithm

Constructive Borel-Ritt interpolation results for functions of several variables

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