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Given a polynomial in several variables is it possible to split it so that all the coefficients of the monomials are +1 or -1.

Example:

p:=-z+2*x+4*y-3*x*y.

I would lie to obtain

f:=-z +x+x +y+y+y+y -x*y-x*y-x*y.

Given a polynomial expression I would like to obtain a list whose entries are the positive entries of the polynomial with multiplicity given by the coefficients:

Example:

Given the polynomial expression: p:=x^2*y-2*y*z+3*x^2+2*y-z

The positive terms are: x^2*y, 3*x^2, 2*y

Thus I would like to obtain the list L:=[x^2*y , x^2 , x^2 , x^2, y , y].

Notice x^2*y appears once since the coefficient is 1.

x^2 appears three times since the coefficient is 3.

y appears two times since the coefficient is 2.

 

 

 

 

 

 

I want to test linearly dependence of a polynomial f on a list of polynomials F by additional condition on parametric coefficients of linear parametric polynomial (linear for variables not parameters). Please note that:

  1. The polynomialand the members of are always homogenous in the variables.
  2. The coefficients of f, the coefficients of the members of F are all always polynomials in the parameters or contant and the members of N and W are all always polynomials in the parameters.

 

For example let

and

(a,b,c,d,e,h are parameters and A1,A2,A3 are variables).

If I use PolyLinearCombo(F,f,{A1,A2,A3}) (see http://www.mapleprimes.com/questions/204469-How-Can-I-Find-The-Coefficients-Of-Linear#comment217621)then its output is false,[].

Now we let to condition sets for parameters as the following:

N:=[ebc+ahd]

W:=[a,c]

The elements of N must be zero means that ebc+ahd=0

and the elements of W are non-zero that is a<>0 and c <>0.

Let a=b=c=d=h=1 and e=-1. This specialization satisfy in the above condition sets N and W. By this specialization we have:

and

Now if I use PolyLinearCombo(F,f,{A1,A2,A3}) then its output is true,[-1,1].

By this additional two condition sets I have to check that whether f is linearly independent of F or not. How can I do this without specialization? In fact I want an algorithm that its input is (null condition N, not-null condition W, list of polynomials F, a polynomial f, the set of variables) and its output is true and coefficients if f is linearly dependent of F w.r.t. null and not-null conditions N and W, else its output is false.

If the name of new procedure is ExtPolyLinearCombo and 

N:=[ebc+ahd]

W:=[a,c]

I want the output of

ExtPolyLinearCombo(N,W,F,f,{A1,A2,A3}) be true,[coefficients]

Thank you very much in advance.

 

 

At the first note that in this question all polynomials have parametric coefficients. Let F be a list of polynomials and f be a polynomial. I want to convert F and f into a linear homogeneous FF and ff resp. At the first I want to sortvthe monomials appears in F and f  w.r.t. a monomial order T and then replace by the new variables A_i.

For example if

and

(a,b,c are parameters and x,y,z are variables) then I want to convert F and f into FF and ff resp:

please note that the variables appears in F and f are:

where sorted by T=plex(x,y,z). Please note that we consider all constants and alone parameters (4, b-4, c-1) as A9. I want to convert v into

and then F into FF and f into ff.

 

How can I extract the coefficients of all monomials in a multivariate polynomial?

For example if f=ax^2+bxy^3+2 then I want

coeff(f,x^2)=a

coeff(f,xy^3)=b

coeff(f,1/2)=4

coeff(f,1/10)=20 and...

I know the Wronskian command. I want to use this command for detecting linearly dependence or independence of some polynomials. I know that the polynomials  are  linearly independent if the Wronskian is not zero.  Conversely, if the Wronskian vanishes  then the polynomials are linearly dependent. Now I want to know that how can I find the coefficient vector if the polynomials are dependent by Wronskian command?

For example if [f1,f2,f3] be a list of polynomials s.t. a1f1+a2f2+a3f3=0. How can I find a1, a2, a3? 

How  we can decide whether a polynomial f is a linear combination of some polynomials g1,...,gm?

For example if f=x^2+y^2 and g1=y+x^2 , g2=y^2-y then f=g1+g2.

Let I  be a polynomial in K[A][X] s.t. A is a sequence of parameters (coefficients of f in F) and X is a sequence of variables. I want to extract the variables from ideal I.

For example if I=[(a-1)x*y^2-b+x, x-y+x^2-c] s.t. a,b,c are parameters and x,y are variables. I want {x,y} as the output of algorithm.

How can I decide that a polynomial is univariate? I want an algorithm that gives a polynomial and its output be true if f is univariate, and be false otherwise.

Let I=<3x^2+2xy+x, y-xy+3, y^2-2x+4> be a polynomial ideal in K[x,y]. I want to form a matrix M corresponding to this ideal as the following:      

                                 x^2     xy     x      y^2      y      constant

                               -----     ----   ----    ----     ----     ------

                                  [3       2       1       0       0           0]

                             M= [0      -1      0        0       1           3]

                                  [0       0     -2        1       0           4]

 

Please note that in the first, the all monomials appeared in generators of I,  sorted by lexicographic ordering x>y. How can I from matrix M from polynomial I?

 Let M be a matrix with polynomial array f_i's such that any array is in K[a_1,..,a_m][x_1,..,x_n] where a=a_1,,,a_m are sequence of parameters and x=x_1,..,x_n are sequense of variables. Now, I want to extract the coefficients of  f_i that are in K[a_1,a_2,..,a_m]. For example if M=Matrix([[ax-bxy],[cx^2-dy]]) how can I extract the matrix coefficint C=Matrix([[a,-b],[c,-d]])?

Please note that a,b,c,d are parameters and x,y are variables.

Problem: I have two polynomials with arbitrary coefficients. I set them both to 0 and I used the 'map' , as well as 'coeffs' command to make the coefficients equal to 0.

Now for some reason, Maple does not print in order for one of the polynomials and it does for the other.

 

Note that 'order' refers to the coefficients attached to the powers of the variable.

 

Quick example: (this one actually works on Maple, but just not the one I have)

 

e1:= ax + (b + c)x^2 = 0

e2:= (c + d)x + (a + c)x^3 = 0

 

After applying map and coeff, one expects it to output

 

a = 0, (c + d) = 0, (b + c)=0, (a + c) = 0

 

instead I got

 

a = 0, (c + d) = 0, (a + c) = 0,(b + c)=0

 

Here is the problematic file

OutOfOrder.mw

Here we have a very brief introduction to the use of embedded components, but effective for the study of the polynomials in operations and some products made with maple 2015 to strengthen and raise the mathematics today.

 

Operaciones_con_Polinomios.mw

(in spanish)

Atte.

L.AraujoC.

This question is 99% similiar to an previously posted question, but this one has a little twist(s).

 

http://www.mapleprimes.com/questions/200101-Solving-For-Coefficients-Of-Polynomial-Equations

 

Here is the sample problem. Let's say I have the following equations

 

e1:=(a+b+1)x^2 + (a+1)*x^4 = 0;

e2:=(a-b)*x+ (a^2 - 1)*x^3 = 0;

 

One can immediately tell the system is inconsistent because on x^3, we have a = -1,1 but on x and x^2 we have a = -1/2

This is precisely the problem I am facing, my e1 and e2 will eventually get bigger (bigger as in the order of the polynomial will increase) and I keep facing inconsistencies.

However if I could find a way to code it so that I can ask Maple to solve only up to a certain degree it would be great. This will eliminate any inconsistency

 

For example, using the system I have here, the system is consistent up to order 2 (it will always start from constants and up to a certain power). So I will ask Maple to solve the above system up to order 2.

Here is a pseudocode I have been working on.

 

For i = 1..2 //this might be increased, but it will of course be finite and probably won't go past 10.

For j = 0..N


//N is the degree of polynomial

f[i]:= coeff(e[i],x,j);

end

end

 

//the above generates the equations, below will solve it

 

solve( { f[i], i=1..?} ,{a,b})

 

Note that the pseudocode assumes e[i] are expressions and not equations set to 0 (which is what I have). Is there an "un"unapply on maple?

 

edit1: have to sleep  (late where I am! Will check for responses tomorrow) thank you

Here we have an application to understand how algebraic expressions, calculating degrees relative abosulutos polynomial operations and introduction to work.Here we have an application to understand how algebraic expressions, calculating degrees relative abosulutos polynomial operations and introduction to work.

 

Grados_de_Polinomios.mw

(in spanish)

Atte.

L.AraujoC.

 

 

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