@Carl Love 

I am puzzled by the fact that you did not use

(parse@cat)(L[]);    # one character shorter

or
parse(cat(L[]));      # two!

(also faster, because it avoids @).

It is not in your style!   :-)

@Markiyan Hirnyk 

It's not wrong at all. Verify!

What could be wrong in my 1 line posted solution:

g(c,d) >= max( f(1), f(-1)) = max (-1-d, 1+d) = abs(1+d) >= 0

?

@Markiyan Hirnyk 

If you can't understand the convexity stuff, just omit it, it is only for the [0,1] case.

@Markiyan Hirnyk 

You start again without those 5 seconds. Please read my answer once more.

Edit. Your question is solved, isn't it?

@Markiyan Hirnyk 

Before throwing those "empty words" you should think 5 seconds:

max{ f(x) : x in [-1,1]} >= max{ f(x) : x in {-1,1} } = abs(1+d).

OK, I simply took the max for x in [0,1] instead of [-1,1] as you asked.

But for x in [-1,1] the problem is much simpler!

g(c,d) >= max( f(1), f(-1)) = max (-1-d, 1+d) = abs(1+d) >= 0

so, in this case g>=0 and is obviously bounded from below.

P.S. I find the problem for x in [0,1] nicer!   :-)

@Markiyan Hirnyk 

Ok, but what's wrong with

add8(275,575);  #   = 1072

(just an example with some of your numbers).

@brian bovril

So, have you tested it? Or, maybe, you are not interested.

@toandhsp

@Christopher2222 

It seems that plot3d produces only square plots (i.e. hight=width).

Ex.

plotsetup(png,plotoutput="d:/tmp/x3.png",plotoptions="hight=1000,width=2500");
plot3d(100*(x^2+y^2), x = -1 .. 1, y = -1 .. 1, scaling=unconstrained);

==> 2500 x 2500 pixels.

In Maple 2016 is it solved?

It is embarrassing that for 3d plots a Windows user (I don'd know the situation in other OS) must use Maxima!

Then simply replace in the code:

if s<>[]

with

if nops(s)=2

@toandhsp 

@toandhsp 

What solutions?
As I understand the problem, one must find the parameters for which the equation has at least one integer solution.

Probably such a general procedure would be useless, the answer being potentially extremely complex.

Even for your very simple example, the answer should be:

If   (l1=0 or theta(t)=Pi/2+_Z1*Pi ) and (l2=0  or beta(t) = _Z2*Pi)  then Rank=1
Otherwise Rank=2.

@cuongtd 

Not quite.

1. A single equation in 3 variables may have exactly 2 solutions. E.g.  x^2 + y^2 + z^2*(z-1)^2 = 0.
2. Inequalities do count! E.g.  x+y = 0, x >= 0, y >= 0.

If you are interested in the posted system (and not in a very general and fast method, which actually does not exist), why don't you use first f7? It simplifies the task a lot as I have mentioned.

BTW, a general method here would de to define a procedure

t1 -> fsolve(...,[p1,t2,p2,t3,p3,x,y])

The problem is that fsolve gives only one solution. But even if you can obtain all the solutions (using e.g. the avoid option, or DirectSearch) it will be very difficult to identify the curves (just think that implicitplot3d usually has difficulties, and in your problem there are 8 variables instead of 3!).

@acer 

Probably a more constructive question would be:

Given a polynomial f in one or more variables find (if possible) polynomials u,v such that
f = a*u^m + b*v^n,
a,b being constants and m,n integers.

@Thomas Richard 

Unfortunately, the .eps for 3d graphics is terrible:

plotsetup(cps,plotoutput=`d:/temp/myplot3.eps`,plotoptions=`landscape`):
plot3d(x^2-y^2,x=-1..1,y=-1..1);

Here is a printscreen (Maple 2015, 64bit, Windows 7):