I recently corresponded with maplesoft on whether the program Groebner:-Basis always produces reduced Groebner bases or not. They say it does. This mw appears to show it producing a non reduced Groebner Basis for a set of polynomials.

More specifically, the coefficient of the lead term of the first polynomial generated is not 1.

I'd like to be shown wrong here, but I am struggling to see what i could be doing wrong.

When I use the Groebner package I normally use it something like this:

with(Groebner):
F≔[x3−3xy,x2y−2y2+x]
Basis(F,plex(x,y))

Does this calculate a reduced Groebner basis or just a Groebner basis? The output looks reduced to me, and I'm sure that I've seen people treat it is as a reduced GB; but the help page doesn't confirm this. Interestingly i can't seem to find any other mention of reduced bases in the help documentation - which makes me think that this method does it (because otherwise people would want a seperate algorithm specifically for getting reduced GBs)

I have an expression involving a sqrt, and i'd like to simplify it, but simplify(expression,sqrt) doesn't seem to fully help.

Eq1 := [(1/2)*(R[b]*kh[a2]+R[m]*kh[a2]-Rh[m]*kh[a2]+sqrt(R[b]^2*kh[a2]^2+2*R[b]*R[m]*kh[a2]^2-2*R[b]*Rh[m]*kh[a2]^2+R[m]^2*kh[a2]^2-2*R[m]*Rh[m]*kh[a2]^2+Rh[m]^2*kh[a2]^2))/kh[a2] = 0, 0 = 0, -(1/2)*(R[b]*kh[a2]+R[m]*kh[a2]-Rh[m]*kh[a2]+sqrt(R[b]^2*kh[a2]^2+2*R[b]*R[m]*kh[a2]^2-2*R[b]*Rh[m]*kh[a2]^2+R[m]^2*kh[a2]^2-2*R[m]*Rh[m]*kh[a2]^2+Rh[m]^2*kh[a2]^2))/kh[a2] = 0]

simplify(Eq1, sqrt)

gives

[(1/2)*(R[b]*kh[a2]+R[m]*kh[a2]-Rh[m]*kh[a2]+sqrt(kh[a2]^2*(R[b]+R[m]-Rh[m])^2))/kh[a2] = 0, 0 = 0, -(1/2)*(R[b]*kh[a2]+R[m]*kh[a2]-Rh[m]*kh[a2]+sqrt(kh[a2]^2*(R[b]+R[m]-Rh[m])^2))/kh[a2] = 0]

which on paper simplifies to:

[R[b]+R[m]-Rh[m]=0,0=0,R[b]+R[m]-Rh[m]=0]

is there a way to get maple to show this?

[In part I am trying to better understand how to manipulate sqrt expressions in maple]

I have a complicated expression which includes RootOf( a quadratic ) but holds for all x what i'd like to do is turn it into a polynomial in x[1], x[2], x[3] so i can start looking at the monomial coefficients.

k[a1]*((x[1]+x[3])*k[d1]+C[T]*k[m])*(R[b]-x[1]-2*x[2])/((R[b]+R[m]-x[1]-2*x[2]-x[3])*k[a1]+k[m])-k[d1]*x[1]-k[a2]*x[1]*(R[b]-x[1]-2*x[2])+2*k[d2]*x[2] = (-R[b]*k[a2]+2*k[a2]*x[1]+2*k[a2]*x[2])*(k[a1]*kh[m]*((x[1]+x[3])*k[d1]+C[T]*k[m])*(R[b]+R[m]-Rh[m]-x[1]-2*x[2])/(k[m]*((R[b]+R[m]-x[1]-2*x[2]-x[3])*k[a1]*kh[m]/k[m]+kh[m]))-k[d1]*x[1]-kh[a2]*x[1]*(R[b]+R[m]-Rh[m]-x[1]-2*x[2])+2*kh[d2]*x[2])/(2*kh[a2]*RootOf(kh[a2]*_Z^2+(-R[b]*kh[a2]-R[m]*kh[a2]+Rh[m]*kh[a2]+2*kh[a2]*x[2])*_Z-2*k[a2]*x[1]*x[2]-k[a2]*x[1]^2+k[a2]*x[1]*R[b]+2*kh[d2]*x[2]-2*k[d2]*x[2])-R[b]*kh[a2]-R[m]*kh[a2]+Rh[m]*kh[a2]+2*kh[a2]*x[2])+(-2*kh[a2]*RootOf(kh[a2]*_Z^2+(-R[b]*kh[a2]-R[m]*kh[a2]+Rh[m]*kh[a2]+2*kh[a2]*x[2])*_Z-2*k[a2]*x[1]*x[2]-k[a2]*x[1]^2+k[a2]*x[1]*R[b]+2*kh[d2]*x[2]-2*k[d2]*x[2])+2*k[a2]*x[1]+2*k[d2]-2*kh[d2])*(kh[a2]*x[1]*(R[b]+R[m]-Rh[m]-x[1]-2*x[2])-2*kh[d2]*x[2])/(2*kh[a2]*RootOf(kh[a2]*_Z^2+(-R[b]*kh[a2]-R[m]*kh[a2]+Rh[m]*kh[a2]+2*kh[a2]*x[2])*_Z-2*k[a2]*x[1]*x[2]-k[a2]*x[1]^2+k[a2]*x[1]*R[b]+2*kh[d2]*x[2]-2*k[d2]*x[2])-R[b]*kh[a2]-R[m]*kh[a2]+Rh[m]*kh[a2]+2*kh[a2]*x[2])

If this were something like q(x)=p1(x)/sqrt(p2(x)) where p1 and p2 are polynomials and q is a quotient- this would be as simple as making sqrt(p2(x)) the subject and squaring both sides, and then movinbg everything onto one and multiplying out denominators. However RootOf is something I'm not used to manipulating.

Is there anyway of converting this expression to a polynomial using maple commands?

I am working on a problem that involves finding a map lambda(x) which maple stores using RootOf:

LambdaMap := [lambda[1] = RootOf(kh[a2]*_Z^2+(-R[b]*kh[a2]-R[m]*kh[a2]+Rh[m]*kh[a2]+2*kh[a2]*x[2])*_Z-2*k[a2]*x[1]*x[2]-k[a2]*x[1]^2+k[a2]*x[1]*R[b]+2*kh[d2]*x[2]-2*k[d2]*x[2]), lambda[2] = x[2], lambda[3] = x[1]+x[3]-RootOf(kh[a2]*_Z^2+(-R[b]*kh[a2]-R[m]*kh[a2]+Rh[m]*kh[a2]+2*kh[a2]*x[2])*_Z-2*k[a2]*x[1]*x[2]-k[a2]*x[1]^2+k[a2]*x[1]*R[b]+2*kh[d2]*x[2]-2*k[d2]*x[2])]

(as an aside, after a few years of using maple and reading the help page I find RootOf confusing, and I think this is the root of this question, I'd love to get some recomended reading on it in the answers)

I want to use the restriction lambda(0,0,0)=(0,0,0) to find relationships between the parameters (kh,k,Rh,R etc)

I tried:

`~`[`=`](`~`[rhs](subs([x[1] = 0, x[2] = 0, x[3] = 0], LambdaMap)), [0, 0, 0]);
solve(%);

which returned:
[RootOf(kh[a2]*_Z^2+(-R[b]*kh[a2]-R[m]*kh[a2]+Rh[m]*kh[a2])*_Z) = 0, 0 = 0, -RootOf(kh[a2]*_Z^2+(-R[b]*kh[a2]-R[m]*kh[a2]+Rh[m]*kh[a2])*_Z) = 0]
{R[b] = R[b], R[m] = R[m], Rh[m] = Rh[m]}

which seems wrong (both the first and third equations are quadratics sharing roots, the first root is _Z=0, and the second is _Z=-R[b]-R[m]+Rh[m]- i'm not sure how either result in this solve)

Is this the right way to do this? i.e. do solve and subs, work intuitively with RootOf expressions.

I am doing some calculus on lambda next - is there anything i should be mindful of when calculating its Jacobean or using diff?