with(Groebner):

gen_ord_upper := proc(k::integer)
	local eqs, nterms, basis, realroots, posroots, i;

	eqs := [
	  -S + A + B + C + D + x + x,
	  -A + A_C*x^2 + C*A_C*x + B*A_B*x + D*A_D*x,
	  -A_t + B*A_B*x + D*A_D*x,
	  -A_C + C + C*A_C*x + A_B,
	  -A_B + B + B*A_D*x + A_D,
	  -A_D + D + D*A_D*x,
	  -B + A*B_0*x^2 + C*B_0 + D*B_0,
	  -B_0 + A*x^2 + C + D + A*B_0*x^2 + C*B_0 + D*B_0,
	  -C + A_t*x^3 + B*x^3 + D*x,
	  -D + k*x
	];

	nterms := [ D, C, B_1, B_0, B, A_D, A_B, A_C, A_t, A, S ];

	basis := gbasis(eqs, lexdeg(nterms[1..-2], [S]));

	print("gen ord upper basis computed", k);
	realroots := fsolve(discrim(basis[1], S) = 0, x);

	posroots := select(type, [realroots], positive);
	print(posroots);

	1/min(seq(posroots[i], i=1..nops(posroots)));
end:

gen_rpn_upper := proc(k::integer)
        local eqs, nterms, basis, realroots, posroots, i;

	eqs := [
	  -S + A + B + C + D + x + x,
	  -A + A_B*x^2 + C*A_C*x + B*A_B*x + D*A_D*x,
	  -A_t + B*A_B*x + D*A_D*x,
	  -A_C + C + C*A_C*x + A_B,
	  -A_B + B + B*A_D*x + A_D,
	  -A_D + D + D*A_D*x,
	  -B + A*B_0*x + C*B_1*x + D*B_0*x,
	  -B_0 + A + C + D + A*B_0*x + C*B_1*x + D*B_0*x,
	  -B_1 + A + D + A*B_0*x + D*B_0*x,
	  -C + A_t*x + B*x + D*x,
	  -D + k*x
	];

	nterms := [ D, C, B_1, B_0, B, A_D, A_B, A_C, A_t, A, S ];

        basis := gbasis(eqs, lexdeg(nterms[1..-2], [S]));
	print("gen rpn upper basis computed", k);

        realroots := fsolve(discrim(basis[1], S) = 0, x);

        posroots := select(type, [realroots], positive);
	print(posroots);

        1/min(seq(posroots[i], i=1..nops(posroots)));
end:

gen_alph_upper := proc(k::integer)
        local eqs, nterms, basis, realroots, posroots, i;

	eqs := [
	  -S + A + B + C + D,
	  -A + A_B + C*A_C + B*A_B + D*A_D,
	  -A_t + B*A_B + D*A_D,
	  -A_C + C + C*A_C + A_B,
	  -A_B + B + B*A_D + A_D,
	  -A_D + D + D*A_D,
	  -B + A*B_0 + C*B_1 + D*B_0,
	  -B_0 + A + C + D + A*B_0 + C*B_1 + D*B_0,
	  -B_1 + A + D + A*B_0 + D*B_0,
	  -C + A_t + B + D,
	  -D + k*x
	];

	nterms := [ D, C, B_1, B_0, B, A_D, A_B, A_C, A_t, A, S ];

        basis := gbasis(eqs, lexdeg(nterms[1..-2], [S]));
	print("gen alph upper basis computed", k);

        realroots := fsolve(discrim(basis[1], S) = 0, x);

        posroots := select(type, [realroots], positive);
	print(posroots);

        1/min(seq(posroots[i], i=1..nops(posroots)));
end: