Questions on the functionality operating on polynomials

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20
votes
2answers
688 views

Funny behaviour when plotting a polynomial of high degree and large coefficients

I am trying to plot a polynomial of degree 29 on the domain [0,1], with fairly large coefficients: ...
-3
votes
0answers
35 views

Write equation as a multinomial in the variables x and Sin[y] and extract the coeficient of x*Sin[y]^2 [closed]

I need help solving this problem. Thank you. (a*x^2 + b*x*Sin[y] + c*Sin[y])^2 + (a*Sin[y]^2 + b*x)^3 Write the above as a multinomial in the variables x and Sin[y] and extract the coeficient of ...
1
vote
1answer
66 views

How can Mathematica help me to find a real radical expression for roots of this polynomial?‎

The polynomial P(x)=x^‎‏4‏‎-‎‏4‏x^‎‏2‏‎-‎‏2‏x+‎‏1‏‎ has ‎‏4‏‎ real roots (this can be clearly checked by plotting). But solving ...
6
votes
2answers
245 views

Finding the coefficient of a certain power in a generating function

I want to compute $(t^1+\dots +t^5)(t^2+\dots+ t^6)(t^3+\dots +t^9)$ and find the coefficient of $t^{15}$ for example. $t$ is an indeterminate Now in Maple this is simply as ...
2
votes
2answers
57 views

Convert polynomial to Chebyshev

I want to convert a polynomial in "standard form" to Chebyshev form. Here's one way to do it: ...
0
votes
1answer
133 views

Finding solutions of polynomials system

Let $f_1,...,f_n$ be a set of polynomials in $x_1,...,x_n$ with rational coefficients. I need to check whether a system $$f_1=a_1,...,f_n=a_n$$ has a real solution for large enough count of points. ...
4
votes
1answer
84 views

On the definition of the associated Legendre polynomials

Mathematica computes for n = 1,2,...: (-1)^n (LegendreP[n, -1, -3]/Sqrt[2]) -I, -3 I, -11 I, -45 I, -197 I, ... Maple ...
3
votes
1answer
81 views

Rewrite an expression as a sum of $SU(2)$ characters?

I have an expression of the form $$q^{-3/2} t^{-7/2}[4qt^2(t + q t^2) + t^2 (q + t) (1 + q t (1 + q t))],$$ I can factor it and write it as $4((qt)^{1/2} + (qt)^{-1/2}) + (qt+1 + ...
0
votes
0answers
27 views

How to prove that all zeros of the complex polynomial $P(z)$ lie in the closed unit disk $|z| \leqslant 1$? [migrated]

I want to know how to prove that all zeros of the polynomial $P(z)$ lie in the closed unit disk $|z| \leqslant 1$. Where $$P(z)=z^{n+1}+\frac{2(n+1)\cos\alpha}{n+2}z^{n}+\frac{n}{n+2}z^{n-1}+\frac{2 ...
5
votes
1answer
116 views

Speed up MinimalPolynomial

My Mathematica code runs slowly MinimalPolynomial[Sqrt[2] + Sqrt[3]+ Sqrt[5]+ Sqrt[7]+ Sqrt[11]+ Sqrt[13], x] runs slowly, but the Maple version ...
0
votes
2answers
238 views

Integrating polynomial functions over polytopes with an add-on package

There is a Mathematica package to evaluate integrals over polytopes: http://library.wolfram.com/infocenter/Books/3652/ In the documentation (Functions.nb file) I ...
0
votes
2answers
111 views

Find Root Iteration

I have an iterative sum from $k=0$ to $k=n$ where the resulting sum is a polynomial of degree $n$. I want to find the numerical root of this polynomial using FindRoot, starting from $x_0$ where the ...
3
votes
1answer
257 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 ...
0
votes
0answers
65 views

Best way to determine polynomial coefficients in series expansion

I would like to solve the equation $$h'(\boldsymbol{x}_1)\left[B_1\boldsymbol{x}_1+g_1(\boldsymbol{x_1},h(\boldsymbol{x}_1))\right]=B_2h(\boldsymbol{x}_1)+g_2(\boldsymbol{x}_1,h(\boldsymbol{x}_1))$$ ...
0
votes
1answer
53 views

Truncating out higher order polynomials in series of fractions

I have taken sequential partial derivatives of a two variable polynomial fraction, resulting in a very long series of polynomials of form: ...
0
votes
1answer
48 views

PolynomialReduce inconsistent results

I was playing around with gröbner basis and s-polynomials and I fell upon the PolynomialReduce, and I was wondering why it gives different results when I move around the polynomials in its second ...
12
votes
1answer
233 views

ToNumberField won't recognize Root as an explicit algebraic number

In Mathematica 9.0.1, it appears that ToNumberField will not always recognize a Root object as an explicit algebraic number. ...
6
votes
2answers
228 views

NSolve missing solutions in Mathematica 10

Running Mathematica 8.0.4 and 10.0.0 on a Windows 8.1 machine. Processed the same code with both kernels: ...
0
votes
1answer
55 views

Is it possible to plot a second-order curve by its non-canonical equation?

I have this second-order polynom: $$ 6xy+8y^2-12x-26y+11=0 $$ And I need to reduce it to a canonical form of a second-order curve. I solved this, but is it possible to draw a plot of the original ...
4
votes
2answers
100 views

Piecewise Polynomial Interpolation

Given some data pairs $(x_i,y_i)$, with $i=0,...,m$, and a degree $r$, I wish to build a piecewise polynomial function to interpolate these data. That interpolation should be continuous, and, on every ...
0
votes
0answers
58 views

Finding Roots of Non-linear Systems: Rescaling polynomials

I'm trying to get all isolated finite equilibria of a moderate multi-dimensional non-linear system of equations. Particularly I have 9 independent variables and third order at most. It turns out that, ...
3
votes
3answers
144 views

Multivariate Polynomial Manipulation

I have a large homogenous multivariate polynomial in, say, 5 variables $a,b,c,d,e$. As an example take the polynomial $$a^4+2abcd+a^2 b^2+e^4+cde^2.$$ Now I would like to replace $k$-th power of any ...
4
votes
2answers
202 views

Why doesn't Roots work on a certain quartic polynomial equation?

In everything that follows I am using Roots to get the solutions to the equations. I start off with this equation: $$ \frac{A}{3}x^4 - x^2 + 2ax - b^2 = 0$$ Now ...
6
votes
1answer
354 views

How can I plot a Chebyshev spiral?

The Chebyshev polynomials T_n of the first kind are a certain set of orthogonal polynomials. They can be defined by T_n(cos(x))=cos(nx), the first of them are T_0(x) = 1 T_1(x) = x T_2(x) = ...
0
votes
0answers
37 views

How to solve equations over polynomial rings

sorry if my question is very basic but I don't know what to even search to look it up and the only "obvious" places I thought of had nothing. Some background, for whatever context it might provide. ...
-1
votes
2answers
109 views
2
votes
1answer
100 views

NSolve for system of polynomial equations

I would like to find all the solutions to the following system of 8 polynomial equations for the variables w1, w2, w3, w4, w5, w6, w7, w8 ...
1
vote
2answers
118 views

Stopping Mathematica from reordering expressions [duplicate]

I want Mathemtica to stop manipulating my polynomials! I mean, I want the output of Print[3 x + 5 + x^2] to be just $3x+5+x^2$, not $5+3x+x^2$ as Mathematica ...
3
votes
2answers
101 views

Number of complex roots for the system of polynomials

I have a system of algebraic equations. For example: $$ \left\{ \begin{aligned} x^2 y + 2 x y + 2 &= 0,\\ y^3 + 2 x + 1 &= 0. \end{aligned} \right. $$ (In my problem the system contains 16 ...
6
votes
4answers
466 views

How to collect terms with positive powers in polynomial

I am trying to collect all terms with non-negative powers of $x$ in polynomials like $\frac{1}{x^2}\left(a x^2+x^{\pi }+x+z\right)^2$ First expand the polynomial ...
6
votes
8answers
889 views

Defining a function that completes the square given a quadratic polynomial expression

How can I write a function that would complete the square in a quadratic polynomial expression such that, for example, CompleteTheSquare[5 x^2 + 27 x - 5, x] ...
2
votes
2answers
73 views

Rounding only coefficients

does Mathematica have built in functionality to round the coefficients of a polynomial to a certain accuracy. Say, we do Print[0.2134320980x^2+0.0023432x] Can ...
6
votes
5answers
205 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, ...
0
votes
1answer
85 views

Find polynomial equation for set of 3D data

data was given in columns in the order {y, x, z}, with y dependent on x, z ...
1
vote
1answer
52 views

sum polynomial H(x,y) [closed]

I want to write and evaluate an expression something like $$H_1=\sum_{i+j=0}^3 e_{ij}x^iy^j$$ or $$H=\frac{\sum_{i+j=0}^3 e_{ij}x^iy^j}{\sum_{i+j=0}^3 a_{ij}x^iy^j}$$ with correct syntax.
0
votes
2answers
73 views

Symmetric function of the roots of a polynomial

First, I'm a beginner. I can compute the sum of roots with the follwing: Roots[x^7 + 5 x^6 + x^5 + x + 1 == 0, x] Plus @@ (x /. {ToRules[%]}) // Simplify Of ...
1
vote
1answer
57 views

Compose two special power series expansions

I have two functions $A(x), B(x)$, given in a special power series form: $A(x)=1-x^{2}\left(\frac{a}{10}-\sum_{k=1}^{9}b_{k}\left(\frac{(x^{2}-1)}{r}\right)^{k}\right)$ ...
2
votes
2answers
179 views

Generating complete lists of polynomials

I would like to generate a list of all $3$-variable Laurent polynomials with non-negative integer coefficients using a looping construct so that I can, one-by-one, check them for specific ...
4
votes
3answers
294 views

Finding parameters making real part of eigenvalues vanish

I have the following $\;3\times3$ matrix: $\left( \begin{array}{ccc} 0.04 -0.4 b & 0 & 0.04 -0.4 b \\ 0 & -0.08-1.2 b & -0.06-0.9 b \\ 1.04 -0.4 b & 2.08 -0.8 b & 0 ...
33
votes
6answers
4k views

Finding real roots of negative numbers (for example, $\sqrt[3]{-8}$)

Say I want to quickly calculate $\sqrt[3]{-8}$, to which the most obvious solution is $-2$. When I input $\sqrt[3]{-8}$ or Power[-8, 3^-1], Mathematica gives the ...
18
votes
5answers
725 views

How do I reassign canonical ordering of symbols?

I have a big polynomial that evaluates to: $$A^2 e^2 \phi ^- \phi ^++A e \phi ^- \phi ^+ c_{2 w} g_Z+\frac{1}{2} A e g h W^- \phi ^+ +\ll13\gg,$$ which is supposed to represent some terms in the ...
2
votes
3answers
288 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 ...
9
votes
3answers
474 views

How do I get a two-term polynomial with a leading negative sign to display in the correct (i.e. textbook) order?

The first three expressions evaluate as expected and the polynomial is displayed in what I would call "textbook" form. The last expression, however, switches the order of terms. Mathematica employs ...
13
votes
3answers
590 views

How to keep Collect[] result in order?

For example, Collect[(1 + x + Cos[s] x^2)^3, x] gives the result ...
2
votes
0answers
28 views

How to simplify a polynomial and get the results in the order that I want? [duplicate]

How to simplify a polynomial in a order that I want? Assume that I have a polynomial here, for example, $a^3 b^4 c^2$, the order of symbels is the dictionary order, $a>b>c$. But what if I want ...
0
votes
1answer
124 views

Determinant of a square matrix with univariate polynomial entries is not a polynom? [closed]

I have a 15x15 Matrix with all polynomial entries. I want to calculate the determinant of the matrix. To my understanding the determinant should be a (albeit high order) polynom, too. And the paper, I ...
0
votes
0answers
74 views

Roots of characteristic equation of sixth order

I have problem how to localize the roots of the sixth order polynomial given in the form of determinant. Classical Solve gives the solution which are in the long form. Is there way how to present ...
1
vote
1answer
66 views

Reducing a multi-variate polynomial to have terms upto certain degree

I have a polynomial in x and y. I want to keep all the terms with (combined) degree 4. Any pointers will be appreciated. Simple tricks for doing the same with univariate polynomial don't seem to work. ...
0
votes
1answer
199 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 ...