## How to retrieve only the monomials (no coefficient...

Hello

I wonder if there is a function to retrieve only the monomials from a multivariable polynomial in x, y and z. Below is one such polynomail.

theta[2]*x3*x0-theta[8]*x2*x1*x0-theta[2]*x2*x1-theta[6]*theta[2]*x2*x0^2-theta[4]*theta[2]*x2*x0-theta[2]*theta[7]*theta[10]*x1*x0^3+theta[8]*theta[2]*theta[9]*x1*x0^2-theta[2]*theta[5]*theta[10]*x1*x0^2-theta[2]^2*theta[3]*theta[10]*x0^3+theta[6]*theta[2]^2*theta[9]*x0^3+theta[4]*theta[2]^2*theta[9]*x0^2

and the result

[x3*x0,x2*x1*x0,x2*x0^2,x2*x0,x1*x0^3,x1*x0^2,x0^3,x0^2]

Many thanks

Ed

## fsolve returns its own statement...

Hi Everyone,

I have a bunch polynomial systems of equations (all form zero-dimensional ideals, i.e. there is a finite number of complex solutions), and I would like to get a real solution for each of them, if available.

fsolve would be the tool to use. But it lead to some strange behaviour for me. Among some other inputs, the input

fsolve({81*x3^12+72*x3^10-614*x3^9+16*x3^8-384*x3^7+1884*x3^6+480*x3^4-2760*x3^3+1600, 81*x2*x3^11+72*x2*x3^9-452*x2*x3^8+16*x2*x3^7-240*x2*x3^6+980*x2*x3^5+32*x2*x3^4-144*x2*x3^3-800*x2*x3^2-220*x3^3+160*x2^2+480*x2+520, 81*x3^11+72*x3^9-452*x3^8+16*x3^7-240*x3^6+980*x3^5+32*x3^4-144*x3^3-800*x3^2+160*x1+160*x2+480},{x1, x2, x3});

somehow outputs itself, i.e.

fsolve({81*x3^12+72*x3^10-614*x3^9+16*x3^8-384*x3^7+1884*x3^6+480*x3^4-2760*x3^3+1600, 81*x2*x3^11+72*x2*x3^9-452*x2*x3^8+16*x2*x3^7-240*x2*x3^6+980*x2*x3^5+32*x2*x3^4-144*x2*x3^3-800*x2*x3^2-220*x3^3+160*x2^2+480*x2+520, 81*x3^11+72*x3^9-452*x3^8+16*x3^7-240*x3^6+980*x3^5+32*x3^4-144*x3^3-800*x3^2+160*x1+160*x2+480},{x1, x2, x3})

I know that this system has no real solutions, but only complex ones. But wouldn't the expected output then be just nothing (as e.g. "solve" does)?

I am confused by this output. Furthermore, how can I "check" with Maple if the output was a solution? By checking the type? There must be less hacky solutions. Thank you all in advance for your help.

Albert

PS.: When I add the keyword "complex" to the function call, then I receive a complex solution; hence, the syntax at least is correct (if someonw might have doubted that).

## how to make desired polynomials with sum of the po...

hi everyone.how can i write a function or procedure or summation so that i can write down the following polynomial ? i just want to create a set of polynomials which their summation of power ( power of x + power of y ) be less than three or equal to three ? the coefficients priority is not important , for example it is not important that a1 multiplies to x or y , i just want to create this polynomial with some coeeficients. tnx for help

 > restart:
 > a[9]*y^3+x^3*a[8]+x^2*y*a[5]+x*y^2*a[7]+x^2*a[2]+x*y*a[4]+y^2*a[6]+x*a[1]+y*a[3]+a[0];
 (1)
 >

## ordering of monomials...

Dear people in Mapleprimes,

I have a question about the ordering of monomials in a polynomial.

I hope you will help me understand how Maple works about it.

I inputed the polynomial as is written in black below.

Then, the outcome was blue, which ordering I could understand well: total degree ordering where at first

those who have the order of 6 are collected which are 14 x^3*y^3, 6x*y^5, and then the following was those which

have the order of 5: 21*x^5, -35 x^4*y, 9*x^3*y^2,-15*x^2*y^3, ... and so on.

And, among those who have the same order, lexical ordering was done, that is among 14 x^3*y^3, 6x*y^5, one which

came first was the one with the larger degree about x, and among 21*x^5, -35 x^4*y, 9*x^3*y^2,-15*x^2*y^3,

the first was 21*x^5, the second was -35*x^4*y, and so one, which was the ordering following the exponent about x.

And, then, I calculated Factor(polynomial) mod 7, which meaning I know.

Then, the result was 2*(x*y+2)*(3*y^3+x^2+3x*y)y.

I can understand the ordering among x*y and 2 in x*y+2, and that among 3y^3, x^2 and 3x*y in 3y^3+x^2*3x*y.

But, I can't understand why (x*y+2) comes at the first term, with 3 y^3+x^2+3x*y following it, and with y coming last.

Best wishes.

taro

 (1)

 (2)

## elementary expression...

Is it possible that this expression has an elementary one (specifically the dilog's):

Y0:=(1/16)*(s*t*(exp(2*t)*s+exp(4*t)+1)*ln((exp(2*t)*s-(-s^2+1)^(1/2)+1)^16*(1+(-s^2+1)^(1/2))^16/((exp(2*t)*s+(-s^2+1)^(1/2)+1)^16*(1-(-s^2+1)^(1/2))^16))+s^3*t*(exp(4*t)+1)*ln((exp(2*t)*s-(-s^2+1)^(1/2)+1)^8*(1+(-s^2+1)^(1/2))^8/((exp(2*t)*s+(-s^2+1)^(1/2)+1)^8*(1-(-s^2+1)^(1/2))^8))+exp(2*t)*t*ln((exp(2*t)*s-(-s^2+1)^(1/2)+1)^32*(1+(-s^2+1)^(1/2))^32/((exp(2*t)*s+(-s^2+1)^(1/2)+1)^32*(1-(-s^2+1)^(1/2))^32))+4*((exp(4*t)+1)*s+2*exp(2*t))*(s^2+2)*dilog((-exp(2*t)*s+(-s^2+1)^(1/2)-1)/(-1+(-s^2+1)^(1/2)))-4*((exp(4*t)+1)*s+2*exp(2*t))*(s^2+2)*dilog((exp(2*t)*s+(-s^2+1)^(1/2)+1)/(1+(-s^2+1)^(1/2)))+((32*s^2*t+64*t)*exp(2*t)+16*(((t+1/8)*s^2+2*t+2)*exp(4*t)-(5/4)*s*exp(-2*t)-(1/8)*exp(-4*t)*s^2+(5/4)*s*exp(6*t)+(1/8)*s^2*exp(8*t)+(t-1/8)*s^2-2+2*t)*s)*arctanh((exp(2*t)-1)*(-1+s)/((-s^2+1)^(1/2)*(exp(2*t)+1)))+8*(-s^2+1)^(1/2)*((1/8)*s*(exp(4*t)+1)*ln((exp(4*t)*s+2*exp(2*t)+s)^12/s^12)+(1/8)*exp(2*t)*ln((exp(4*t)*s+2*exp(2*t)+s)^24/s^24)+(s^2-6*t-3)*exp(2*t)+((-(1/8)*s^2-3*t)*exp(4*t)+s*exp(-2*t)+(1/8)*exp(-4*t)*s^2+s*exp(6*t)+(1/8)*s^2*exp(8*t)-(1/8)*s^2-3*t)*s))/((s*exp(-2*t)+exp(2*t)*s+2)*(exp(4*t)*s+2*exp(2*t)+s)*((-s^2+1)^(1/2)+2*arctanh((-1+s)/(-s^2+1)^(1/2))))

Also I'm wondering since Y0 should solve the ode

-(diff(diff(y(t), t), t))+(4-12/(1+s*cosh(2*t))+8*(-s^2+1)/(1+s*cosh(2*t))^2)*y(t) = C/(1+s*cosh(2*t))

with some constant C but I only get rubbish.

I ask this because I found that in another context this seems to be correct:

f1:=-(1/12)*Pi^2*((-s^2+1)^(1/2)-arccosh(1/s))/(-s^2+1)^(3/2)+(1/12)*arccosh(1/s)^3/(-s^2+1)^(3/2)-(1/4)*arccosh(1/s)^2/(-s^2+1)

f2:=(1/2)*((-s^2+1)^(1/2)*(polylog(2, s/(-1+(-s^2+1)^(1/2)))+polylog(2, -s/(1+(-s^2+1)^(1/2))))-polylog(3, s/(-1+(-s^2+1)^(1/2)))+polylog(3, -s/(1+(-s^2+1)^(1/2))))/(-s^2+1)^(3/2)

and f1=f2

but maple doesnt convert it.

Also maple has trouble to convert

2*arctanh(sqrt((1-s)/(1+s)))=arccosh(1/s)

everywhere: 0<s<1

## Maple 17: High Performance Polynomials

by: Maple

As described on the help page ?updates,Maple17,Performance, Maple 17 uses a new data structure for polynomials with integer coefficients. Our goal was to improve the performance and parallel speedup of polynomial algorithms that underpin much of the system and create a platform for large scale polynomial computations. Shown below is the new representation for 9xy3z

## Galois group with parameters...

I consider a polynomial \$P(x)\$ such that their coefficients are in \$\mathbb{Q}(u_1,\cdots,u_k)\$ where \$u_1,\cdots,u_k\$ are complex parameters. I use in Maple the command \$galois(p(x),x)\$ and I obtain a fixed Galois group solution.  Fortunately, when I give explicit values (randomly chosen) to the \$(u_i)_i\$, I obtain always the previous group as Galois group. I think that Maple considers that the \$(u_i)_i\$ satisfy no algebraic equations, that is the \$(u_i)_i\$ are generic....

## How do I extract the corresponding variables given...

I have a multivariate polynomial equation, in that somehow I know the coeffcients, using this information, I want to extract the variables. This will be the opposite of coeffs function.

for e.g. I have 3*x3 + 5*x4

Given 3 and 5, I want to extract x3 and x4.

Satya