Mathematical Aspects and Physical

We review the problem of the tuneshift correction of symplectic maps and its applications to accelerator physics. The normal form techniques used to compute the correcting terms are outlined both in two and four dimensional phase space. A theorem which gives an upper bound to the growth of the coeecients of the correcting map is proved. Two ways of correcting the tuneshift are discussed: a compensation obtained by adding multi-polar terms and a procedure based on the composition with a normal form. Both cases are analysed numerically, showing for the quadratic map a factorial divergence in agreement with the proved theorem. Applications of the tuneshift correction to real machines are discussed, referring in particularly to LHC.