Questions on the functionality operating on polynomials

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1
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1answer
96 views

How to transform the polynomial in this way?

I have two functions f and g and a polynomial ...
6
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2answers
294 views

Why does PolynomialQ[x^n, x] return False?

From what I can see PolynomialQ will return False whenever some exponent is another variable such as here: ...
4
votes
1answer
347 views

Factoring a two variable polynomial in a special way

Let $$f=x^9-x^6+4x^5y+2x^3y^2-y^4$$ I would like to factorize $f$ into form: $$(y-F_1(x))\cdots(y-F_k(x))$$ over complex numbers. How can I do it with Mathematica?
6
votes
2answers
273 views

Finding common roots of a specific system of polynomials

Some of my colleges are interested in finding common zeros of the following four polynomials: ...
2
votes
1answer
107 views

How can I sort the polynomial by the degree of $x$

f1 = a x^2 + b x + c + x^3 (* a x^2+b x+c+x^3 *) f1 /. x -> (x - a/3) // Expand (* (2 ...
4
votes
2answers
291 views

Dimension of an algebraic variety

I would like to compute, using Groebner bases, the dimension of the variety defined by a set of polynomials in several variables. In the wikipedia page the method is described, and an implementation ...
2
votes
4answers
160 views

Get the first positive coefficient in polynomial?

I have Polynomial $F(x) = \sum_{i \leq n}{a_ix^i}$. How to get the first $a_i > 0$? Thanks,
2
votes
1answer
216 views

How can I calculate all irreducible polynomials of 31 degree in $\mathbb Z_2[x]$?

How can I calculate all binary irreducible polynomials of degree 31? or how i calculate all irreducible $f$ in $\mathbb Z_2[x]$? (The irreducible polynomial in $\mathbb Z_2[x]$ and $\mathbb R$ are ...
0
votes
1answer
313 views

Partial factorization of multivariate polynomials in terms of given polynomials

I have calculated several homogeneous polynomials in 4,5 or 6 variables $t_1,\dots,t_6$. I would like to rewrite them as a sum of products of specific lower degree polynomials, which have a meaning in ...
2
votes
1answer
595 views

Formatting results of a polynomial long division

I am teaching polynomial long division to my high school students. Not a pleasant topic to have to cover. I went to use Wolfram|Alpha and obviously, internally, they have a really elegant way to ...
3
votes
6answers
461 views

Collecting terms of even exponents

Say I've got a polynomial of $4$ or $5$ variables (we'll say $d_1$, $d_2$, $d_3$, and $d_4$). How would you collect the terms where each $d$ is raised to an even power? It collects the terms of the ...
6
votes
1answer
184 views

The third argument of the Root function

Consider the roots of an arbitrary indecomposable polynomial: sol = Solve[x^5 + 2 x + 1 == 0, x] The returned expression is ...
0
votes
1answer
369 views

Solve[polynomial, x, Reals] doesn't get all real roots or correct ones?

My main confusion is about the difference between the two code blocks at the end of this long spiel, but the spiel contains the code to create the polynomial if its coefficients are helpful for ...
4
votes
4answers
268 views

How to compute MIN$(A(x), B(x))$

I have two polynomials: $\;A(x) = \sum a_i\;x^i$, $\quad B(x) = \sum b_i\;x^i$. Given $A(x)$, $B(x)$, I want to compute $\;C(x) = $MIN$(A(x), B(x)) = c_i\;x^i$ where $c_i = $MIN$(a_i, b_i)$. How can ...
4
votes
2answers
180 views

HornerForm of polynomials in terms of E^(i x)

I want to know how to get the HornerForm of the following expression in terms of E^(I x): ...
12
votes
4answers
374 views

How to replace every possible $A+B$ and $AB$ in expansion of $(A+B)^{10}-A^{10}-B^{10}$ with $x$ and $y$, respectively?

I would like to replace every possible $A+B$ and $AB$ in expansion of $(A+B)^{10}-A^{10}-B^{10}$ with $x$ and $y$, respectively. How to do it with the simplest code in Mathematica? For example, ...
5
votes
2answers
437 views

Adapting CoefficientList (and the related functions) to work with Laurent polynomials

Is there a slick way to make CoefficientList (and the other similar functions, CoefficientRules etc) work for Laurent polynomials (i.e. where negative exponents can occur), if I don't know a priori ...
1
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2answers
440 views

Approximate solution to system of polynomial equations

I'd like to solve approximately the following system of equations for $a_1$, $a_2$, $\alpha_1$ and $\alpha_2$ all real and nonzero: ...
4
votes
1answer
582 views

Implementation of the Polynomial Chinese Remainder Theorem

I would like an implementation of the Chinese Remainder Theorem for polynomials in $\mathbb{Z}[x]$, that is, a function ...
5
votes
2answers
595 views

How to find monotonically increasing intervals of a function

I tried this code, but not working Clear[f]; f[x_] := x^3 - 3 x + 2; ForAll[{x1, x2}, x1 < x2, f[x1] < f[x2]] Reduce[%, {x1, x2}, Reals] I expected the ...
0
votes
0answers
236 views

symbolic solution for a system of nonlinear equations in mathematica [closed]

Would appreciate any help on the following in mathematica I cant figure it out. I have a system of equations that I am trying to solve symbolically. I have 9 equations and 8 unknowns (I have also ...
4
votes
1answer
130 views

How to check if a Polynomial has a specific form

I have a polynomial F[x], for example F[x] = 1 - 2x + x^2. I wanna check whether F[x] has the form of ...
2
votes
1answer
218 views

Issue with Coefficient command [closed]

I'm trying to used the Coefficient command to extract the numerical values in front of a Chebyshev polynomial. I know that there is a numerical way to do this, presented in numerical recipes, which I ...
7
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1answer
376 views
3
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1answer
392 views

Transform recursion for coefficients into differential equation for generating function

Assume, one is given a linear recursion with polynomial coefficients for a sequence $(a_i)_i$, such as a[i] == i a[i-1] I would like to convert this recursion ...
1
vote
0answers
258 views

Is there a way to speed up Simplify and/or PolynomialReduce Modulus-> 2?

I'm trying to simplify a series of equations with at most 64 input terms. As the number of terms involved in the equations increase, the runtime seems to grow exponentially. Does anyone know of ways ...
3
votes
1answer
256 views

Symbolic Integration of Special Functions

Sorry in advance if this formatting comes out strange, this is my first question! I can't find a way to integrate, e.g., a function of the Hermite polynomials for general (still integer) order. For ...
2
votes
1answer
433 views

How to get the coefficient list

polynomial=-x^4+2 b x^3+(b^2-c^2+2 c) x^2+(2 b c-2 c d) x+c^2-d^2 This is good CoefficientList[polynomial, x] But how to ...
3
votes
3answers
436 views

How can I reorder the factors in the terms of a polynomial?

How can I reorder the factors in the terms of a polynomial? Consider ...
0
votes
1answer
264 views

Is the formula $\sum _{m=1}^{n-1} \prod _{k=m+1}^n x_k x_m$ wrong in the wiki's page

SymmetricPolynomial[2, {Subscript[x, 1], Subscript[x, 2], Subscript[x, 3], Subscript[x, 4]}] $$\begin{align*}x_1 x_2+x_3 x_2+x_4 x_2+x_1 x_3+x_1 x_4+x_3 ...
1
vote
2answers
94 views

The order of the result $x^2 \left(b-\frac{a}{2}\right)+(a-2) x^3+\left(2-\frac{b}{2}\right) x+4 x^4-1$ [duplicate]

Can you explain me a little how Mathematica sort this result, and how to sort in descending powers of $x$ ...
0
votes
2answers
120 views

Rearranging a simple algebraic equation

Suppose I have a simple algebraic equation like: ChebyshevT[4, p] == 0 1 - 8 p^2 + 8 p^4 == 0 and I want to solve for the ...
1
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2answers
408 views

How to know form of plotted Bézier function

Simple scenario is to see the Bézier function, but how to know which polynomial approximate it? ...
5
votes
2answers
308 views

Implementation of a complex recurrence relation for polynomials

I would like to implement the recurrence relation for the polynomials $U_n(x)$ appearing in the large order asymptotics of the Bessel functions. The recurrence in question is: ...
0
votes
1answer
1k views

How to neglect higher power terms in a polynomial expression

I have a polynomial expression of order n (say n=20). F(x)=1+x+x^2+x^3++...x^20. I want to approximate the polynomial for order 3 only. So I need to make the ...
6
votes
5answers
229 views

Lowering the degree of an polynomial with an assumption that the polynomial has a factor x^2+ax+b

Let's assume $ x^{10}+x^5+1 $ has a factor $x^2+ax+b$ Then, If $x^2=-ax-b$, $ x^{10}+x^5+1 =0$. If we successively apply $x^2=-ax-b$ to $ x^{10}+x^5+1 $, we can make $ x^{10}+x^5+1 $ to degree 1, ...
1
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0answers
377 views

Polynomial factorization over finite fields with non-prime order

One can easily factor a polynomial over finite fields of prime order, using Factor command: ...
0
votes
1answer
673 views

Factoring a quintic

I am trying to prove that a quintic polynomial, $p(x) = A5 x^5 + A4 x^4 + A3 x^3 + A2 x^2 + A1 x + A0$, which admits at most three real roots. Unfortunately Descartes' rule of signs does not help, ...
3
votes
3answers
363 views

Write a function using pattern test to test whether the input is polynomial and print error message if not

I am doing this which is supposed to be an easy problem and which I think my code should be correct but for some reasons it is not working. May anyone help correcting my mistakes? How should I write ...
5
votes
1answer
136 views

Unexpected result for Coefficient[]

Is this just me being stupid, or is this a known bug in Mathematica? Coefficient[2 x + 2 y, x + y] gives 0, while ...
7
votes
1answer
797 views

Writing an expression as sum of squares of expressions

Suppose we have a symmetric homogeneous polynomial expression $P$ in $X=(x_1,\cdots, x_n)$. I want to check whether there are functions $g(X)$ so that $P$ is of the form $\sum _{1\le i<j\le n} ...
0
votes
1answer
119 views

Solving one equation, then inputting the values into another for NonlinearModelFit

Considering the equation a*x^3 + a*x^2 + x + b == 0 I'm looking to find the best value for a and ...
5
votes
2answers
1k views

Inverse of a polynomial in a polynomial ring

Let $N$ be a prime, and $q$ be a positive integer. Given a polynomial $f(x)$ in $R = \mathbb Z[x]/(x^N-1)$, I want to find another polynomial $f_q(x)$ in $R_q = \mathbb Z_q[x]/(x^N-1)$, such that ...
1
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3answers
385 views

List of Tribonacci Polynomials with Mathematica? [duplicate]

I want to list top ten of Tribonacci polynomials. I have following algorithm, but it doesnt work. ...
1
vote
3answers
2k views

The plot of roots of polynomials

I have polynomial equation like Tribonacci Polynomials for example: $T_3(x)=x^4+x$. After finding the roots of this polynomial, I want to show these roots in the complex plane. I have tried lots of ...
15
votes
2answers
336 views

What are Root objects with multiple polynomials?

In Mathematica 9 a new flavor of Root object with multiple polynomials was introduced. For example, ...
7
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2answers
2k views

How to define a polynomial/function from an array of coefficients?

I have the coefficients of my desired polynomial in an array CoefArr (I'm new to mathematica, so I think of everything as arrays, it is actually a list I believe) starting with the constant at index ...
2
votes
3answers
329 views

Symbolic manipulation of functional form

I have a functional polynomial expression of the form: ...
2
votes
2answers
345 views

Evaluating Polynomials at Grid Points

I am continuing my quest on B-splines. The code below builds a 5x5 matrix out of B-splines, using the BSplineBasis[] routine. I now want to evaluate the polynomials that are stored in each matrix ...
13
votes
1answer
334 views

ToNumberField won't recognize Root as an explicit algebraic number

Bug fixed in 10.0.0 In Mathematica 9.0.1, it appears that ToNumberField will not always recognize a Root object as an ...