## Section: New Results

Keywords : polynomial approximation, minimax approximation, Chebyshev polynomials, automatic generation, floating-point arithmetic, polytopes, linear programming, modular function, modular invariant, Fourier coefficient, circle method, Hankel function.

### Approximation

Participants : N. Brisebarre, J.-M. Muller, A. Tisserand.

#### Efficient Polynomial Approximation

In [1] , N. Brisebarre, J.-M. Muller and A. Tisserand provide a general and efficient method for finding the best polynomial approximation under constraints of form and size in bits of the coefficients. The method described in [1] is currently implemented in the C library MEPLib (cf. § 5.10 ).

#### Modular Functions

The modular invariant
jis an important function in number theory
[64] . G. Philibert (LARAL, Saint-Étienne) and N. Brisebarre established in
[16] precise upper and lower bounds for the Fourier coefficients of
j^{m} for all
. These results improve on previously known results, especially those of Mahler
[60] and Herrmann
[57] .