@Axel Vogt   

Here are 2 reasons for the sharp decrease in the number of solutions:
1) All variables are different.
2) The ranges for all variables are significantly narrowed.

@MiguelJesus   From the inequality  18-J-K>0  we get  J+K<18 . From this we get the bounds  1<=J<=17  and  1<=K<=17 .

Next from the inequality  33-K-L>0  we get  K+L<33  and because bounds for  K are already known, then we get  1<=L<=32 .  The bounds for  H  and  F  we get from the inequalities  34-H-J-L > 0  and  -15-F+J+K+L > 0 

For  F  we have   F< J+K+L-15 < J+K+K+L-15 < 18+33-15=36 

@Markiyan Hirnyk   For example, if necessary to find the total number of solutions.

@Markiyan Hirnyk   My method gives the complete list of all solutions in explicit form. In some situations, it may be important!

@Markiyan Hirnyk   In Maple 12, we immediately obtain the result

RealDomain[solve](r^6+r^4+r^2-10 = 0);

In Maple 16 additional command is needed:

RealDomain[solve](r^6+r^4+r^2-10 = 0);

allvalues([%]);

@Voltageist   Look closely at the output. You can see the exact answer to your problem:

@Preben Alsholm  Thanks for improvement. Another equivalent variant:

H:=G@@2;

@Markiyan Hirnyk   you're right! Cases  a=0  or  h=0  or  k=0  require special consideration.

@John 2014  Your problem is reduced to solving a system of linear equations (see the above link).

Solution for the base L1 (the remaining items can be solved similarly):

Poly:=x*(t^2+1)+y*(t-2)+z*(t+3);  Poly2:=a1*t^2+ a2*t+ a3;

Sys:={seq(coeff(Poly,t,n)=coeff(Poly2,t,n), n=0..2)}; #  The system of linear equations

solve(Sys, {x,y,z});

1) Use Maple syntax for imputs of formulas, for example,  a1*t^2+a2*t+a3 . Your formulas are incomprehensible.

2) See  the question

• Showing that a list is an ordered basis for a subspace

@mahmood180   Replace  Int  by  int

@Markiyan Hirnyk 

execution.mws

Carl's code is working:

Sol:=fsolve({F,Fw,Fk,Ft}, map(`=`,{w,k,ki,T},0..infinity));
evalf(eval({F,Fw,Fk,Ft}, Sol));

@apm   Since  limit  is weaker than  MultiSeries:-limit  then before  limit  you have to use a simplifying transformation as shown by Axel Vogt. In addition, instead of  combine, you can use  expand  command:

restart;

Expr:=sqrt(-2*cos(alpha)*cos(alpha+d)+2-2*sin(alpha+d)*sin(alpha))/d:

limit(expand(Expr), d = 0, left);  limit(expand(Expr), d = 0, right);

                                                         -1

                                                          1

@Markiyan Hirnyk  Ill-conditioned and ill-posed systems of linear algebraic equations are the different things.