Specificity for the analysis of parametric polynomials

The following variables play a role in the procedures involving a parametric polynomial:

• `ALIAS/opt_min`, `ALIAS/opt_max`: initial value for the minimum and maximum real roots of a parametric polynomial
• `ALIAS/stop_opt_sol`: if set to 1 the algorithm will exit soon as a maximum greater than
  `ALIAS/opt_sol_max` or a minimum lower than `ALIAS/opt_sol_min` has been found. If set to 2 while looking for a minimum and a maximum the algorithm will exit if both the minimum is lower than `ALIAS/opt_sol_min` and the maximum is greater than `ALIAS/opt_sol_max`

For the evaluation of the coefficient of the polynomial there may be evaluation problem (for example one coefficient involves a division by an expression whose interval evaluation may include 0). To deal with this case a procedure similar to what is done for MakeF can be used:

• the variable `ALIAS/user_CoeffINIT` allows to define auxiliary variable in the C++ procedure that implement the evaluation of the coefficients of the polynomial
• a procedure ALIAS_Coeff(fid,i) may be created that determine if the coefficient i may be evaluated and if not affect to him an arbitrary large value.

To create the procedure ALIAS_Coeff we may also use the Problem_Expression package (see section 2.1.5). The procedure ALIAS_Coeff will be obtained from a list of coefficients Coeff in the variable Vars with the following maple code:

 
`ALIAS/low_value_expr_violated`:= -1e20;
`ALIAS/high_value_expr_violated`:= 1e20;
Verify_Problem_Expression(Coeff,Vars,"ALIAS_Coeff","ALIAS_Coeff");

Finally the above procedures may use the simplification procedure BoundUP, see section 8.1.1.1, that uses general roots bound algorithms for determining if a polynomial may have a root within a given interval.