## 622 Reputation

14 years, 260 days

## Characteristic....

@Carl Love I have forgotten about this option completely, my apology. Unfortunately I have no access to the Groebner package. I can only suggest using method=buchberger and plex(X,Q). Quote from the help file for GB:

- GB is an implementation of Buchberger's algorithm for computing Grobner bases
over Zp(a,b,...)[x1, ..., xn].

- GB(F,X,termorder) computes the reduced, minimal Grobner basis of the
polynomials F with respect to the indeterminates X and the given term order-
ing.

No help otherwise.

..Also the basis you quoted is identical to tdeg basis.

## Monic....

Your definitions for x and q are:

 L := 1+X+X^6+X^7+X^8, (X^3+X+1)*Q+Q^3+(X^7+X^6+X^4+1)*Q^2+X^7+X^6+X^4+X^3+X^2+X+1;

so the monic for Q is what you seek and it is obtained with:

 GB([L], [X, Q], plex) mod 2;

## Excellent reference....

@roman_pearce Thank You, that is precisely the function I needed. But also I was wondering if there is any support for this from side of Maple (even though the solution is so simple).

## Do you mean this:...

 L:=[seq([seq(x^i*y^j,j=1..5)],i=1..5)];[[op(-1,op(-1,L)),op(1..-2,op(1,L))],seq([op(-1,op(i-1,L)),op(1..-2,op(i,L))],i=2..nops(L))];;

## Do you mean for this to be a question or...

Perhaps you want to try it using symmetric polynomials:

 A := [(a+b+c+d)*(a*b+a*c+a*d+b*c+b*d+c*d)-12*sqrt((a^2+b^2+c^2+d^2)*a*b*c*d), [a+b+c+d-1]];G := [-a*b*c*d+p14, -a^2*b-a^2*c-a^2*d-a*b^2-a*b*c-a*b*d-a*c^2-a*c*d-a*d^2-b^2*c-b^2*d-b*c^2-b*c*d-b*d^2-c^2*d-c*d^2+p23, -a^2-a*b-a*c-a*d-b^2-b*c-b*d-c^2-c*d-d^2+p22, -a-b-c-d+p11], [a, b, c, d, p11, p22, p23, p14];simplify(A,G);simplify(A,map(op,[G[1],A[2]]),G[2]);

Look for symmetric polynomials package.

## Small....

@Carl Love I was supposing this could work:

 R := {Product(g-i, i=-4..-3), Product(f-i, i=0..2)};#R := {product(g-i, i=-4..-3), product(f-i, i=0..2)};

But now I think it would work only for small cases.

## That was Maple V....

@Carl Love Maple 16 seems nicer.

What would this produce, 25 answers?

 isolve({x>3/2, x<13/2, y>3/2, y<13/2});

Also

 S := {-1/2 < 2/53*f+7/53*g, 7/53*f-2/53*g < 1/2, 2/53*f+7/53*g < -37/106, 3/106 < 7/53*f-2/53*g}; R := {(g+4)*(g+3), f*(f-1)*(f-2)};isolve(S union R);

gives full solution. So a solution would be to first figure each variable's range. In observation it seems a false approach for isolve to produce over a cartesian product of ranges.

Does isolve accept inert Product expressions?

## Divide....

Dividing (c,d) plane with the polynomials involved this image is obtained. The range of x: -1..1 was not used.

@Carl Love The order of eliminations I employed:

{cos(p1), sin(p1), sin(p2), cos(p3), sin(p3), cos(p2)}, {cos(t3), sin(t3)}, {sin(t2)}, {cos(t2)}, {y}.

Eliminating cos(t2) gave the first occurence of RootOf. Eliminating y gave 3 cases, all quadratic in x. That is my previous post. Perhaps you could reproduce this.

## Sequential elimination....

@cuongtd Eliminating sequentially individual variables reduces the system to this:

 proc()local t264, t334, t309, t310, t336, t312, result, t337, t292, t339, t343, t321, t344, t346, t345, t323, t299, t347, t304, t324, t342, t325, t328, t307,t326, t332, t331, t333, t306;global x, y, eta1, eta3, eta2, Alpha1, Alpha2;    t328 := Alpha2*eta3;    t325 := 8*t328;    t347 := t325 + 4;    t326 := -4*t328;    t333 := x*eta2;    t346 := t326 - 4*t333;    t309 := eta2^2;    t310 := eta3^2;    t304 := Alpha1^2;    t336 := 4*t304;    t306 := Alpha2^2;    t345 := (-8*(Alpha1 - eta2)*x - 4*t310 - 4*t306 + t336 - 4*t309 + t347)*eta1;    t332 := Alpha1*x;    t292 := -4*t332;    t344 := t292 + 4*t333 - 2*t309;    t312 := x^2;    t299 := 2*t304;    t307 := eta1^2;    t343 := 2*t309 + 4*t312 - 4*t307 - 2 + t299 + t292 + t346;    t331 := t304 + t306;    t342 := -t331 + 2*t332;    t339 := -8*t312;    t337 := 4*t309;    t334 := - 1/2*1/(-eta3 + Alpha2);    t324 := x*t325;    t323 := -1 + t309 + t310 - 2*t333 + t342;    t321 := 1 + t332*t337 + t299 + Alpha1*t324 + t347*t307 + (t339 + 4)*t328 + (Alpha1*t339 + t324)*eta2 + (t336 + t337)*t312 + (t309 + t346)*t309         + (t326 + t304 + t344)*t304 + (t306 + t343)*t306 + (6*t306 + t310 + t343)*t310 + t344;    t264 := -1 + t331;    result := [[{y = (2*eta1 + t323)*t334}, {t264, t321 - t345}], [{y = eta3}, {t264, -2*t328 + t310 + t312 - t307 - t342}],        [{y = (-2*eta1 + t323)*t334}, {t264, t321 + t345}]]end proc(), {Alpha1 = cos(t1), eta3 = .995, eta2 = .575, eta1 = 1.35, Alpha2 = sin(t1)};

The above and the below is from the previous post.
Top view:

Most likely the elimination ran into trouble and substituted lesser cases(?).

## No curves?...

@cuongtd Well, what I've found was done by elimination. Oddly enough, when I converted to pure symbolic and solved again I received a solution that has NO curves, and has only points termed in the constants of the equations. The following list has complex values removed:

[x, y, cos(t1), sin(t1)] = ([1.15, -.35, .5289398012, .8486593467], [1.15, -.35, -.3352457626e-1, -.9994378934], [1.15, .35, -.3352457626e-1, .9994378934], [1.15, .35, .5289398012, -.8486593467], [0, -.35, 0, 1.], [0, .35, 0, -1.], [0, -2.35, 0, -1.], [0, 2.35, 0, 1.], [.575, 1.345, -.6440493667, .7649839300], [.575, 1.345, .9980143844, .6298641560e-1], [.575, -1.355, .9973489646, -.7276704453e-1], [.575, -1.355, -.6406294899, -.7678501525], [.575, .645, -.7749377670, .6320375442], [.575, .645, .7165251908, -.6975612166])

## Compose...

Thomas Richard already covered this, but you might lke these:

f := () -> `@`(parse, cat, op, map)(convert, [args], string);
g := (L::{list, set}) -> `@`(parse, cat, op, map)(convert, L, string);

f(100.100, 3, 3, 2, 2, 1, 1, `535`);
g([100.100, 3, 3, 2, 2, 1, 1]);

## Suspicion....

@Carl Love I was suspecting that by now Maplesoft developed a feature for this.

## More digits....

@Majmaj Since precision is your issue the last line should be with increased Digits. Pick the appropriate setting; I used 2*Digits:

moreDigits:=2*Digits;
S:=evalf(`@`(f,op,map2)(op,1,B),moreDigits),`@`(op,map2)(op,3,B);

evalf((minimize..maximize)(S),moreDigits);

or in one line:

moreDigits:=2*Digits;
evalf((minimize..maximize)(`@`(f,op,map2)(op,1,B),`@`(op,map2)(op,3,B)),moreDigits);