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Questions tagged [polynomials]

Questions on the functionality operating on polynomials

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0 votes
1 answer
49 views

Finding kernel of a set of polynomial functions

I have a list of polynomials fi[x_] and polynomial fractions qi[x_] (with i as an iterator) ...
7 votes
4 answers
342 views

PolynomialQ behaviour

I am crafting some functions on polynomials that must be in x. But I checked that it is always True whatever variable I use: ...
3 votes
1 answer
100 views

How to `Collect` where `var` is one expression?

Edit: Clear["Global`*"]; expr1 = (a + c - b + 1)^3 + (-b + a + c + 2)^4 // Expand For test case expr1, this is the ...
0 votes
1 answer
39 views

Extracting matrix coefficients from multivariate matrices

Suppose we have a multivariate polynomial matrix $P(\mathbf x) = \sum_\alpha A_\alpha\mathbf x^\alpha$ (in multi-index notation) where the $A_\alpha$ are constant coefficient matrices. I am looking ...
5 votes
1 answer
190 views

Finding a rational univariate representation (RUR) for a polynomial system

According to Wikipedia, a RUR of a zero-dimensional system consists in a linear combination of the variables, $x_0$ called ''separating variable'', and a system of equations : \begin{cases} h(x_0)=0\\...
2 votes
2 answers
79 views

Unstable work of PolynomialMod

I tried to use PolynomialMod for my calculation and i need some help on working with it, because i'll need to use it several hundred times. I made such a request ...
1 vote
3 answers
97 views

How to sort the terms in a polynomial according to their degree?

I have the polynomial ...
0 votes
1 answer
125 views

How to execute this code? [closed]

I want to plot the following equation: ...
1 vote
0 answers
56 views

Replacement rule runs for a long time [duplicate]

I have a simple expression that reads (some expression in b) q^(8/3): ...
1 vote
1 answer
56 views

How to get the multiplicity of a certain root in a system of polynomial equation

Suppose I have a system of polynomial equations with base field $\mathbb{C}$ with $n$ equations and $n$ unknowns, how can I get the multiplicity of a certain solution? As an example, consider $$x^3=0\\...
0 votes
0 answers
34 views

How to prevent factorization of polynomials in output

The output of my program is a matrix with polynomial entries. Most often Mathematica rearranges the polynomials with some partial or full factorization, as in $p^2q+q^2(p+q)$ or $p(p+q)$. I would like ...
3 votes
3 answers
1k views

How to find the lowest power of variable in expression?

If I have expression like a1/x +a2/x^2 + a3/x^3 I want to return 1/x^3. In general case, ...
5 votes
4 answers
290 views

How to convert a polynomial into a list?

I have a list of polynomials. Each polynomial looks like: p[1, 2, 5] p[3, 6, 9] p[4, 7, 8] - 3 p[1, 2, 4] p[3, 6, 9] p[5, 7, 8] How to convert each of these ...
5 votes
3 answers
370 views

How to get the analytical form of a solution to an algebraic equation?

The real analytical solution of the algebraic equation $x^5 + 10 x^3 + 20 x == 4$ is $x=-2^{2/5} + 2^{3/5}$, how to get it with Mathematica? I've tried with ...
0 votes
3 answers
91 views

How to find the maximum value of coefficient k that satisfies the condition?

The known equation is: (1 + 62 x)^99 + (62 - x)^99 == a0 + a1 x + a2 x^2 + a3 x^3 + ... + a99 x^99 And ...
3 votes
2 answers
138 views

How to divide a known root out of a polynomial?

Consider this simple 5th order polynomial: pol=x^5+x^4+1 Factor[pol] (* (1+x+x^2)(1-x+x^3) *) root=FindInstance[pol==0, x] (* -1/2 -I/2 Sqrt[3] *) ...
1 vote
2 answers
84 views

Ordering of solutions of NSolve

I want to compute the solution to an equation with a parameter. It turns out that this solution is a sextic polynomial and mathematica outputs the solutions in a root form. Here's my code ...
1 vote
0 answers
68 views

Problems with implementation of subresultant pseudo-remainder sequence in Sturm's theorem

I've implemented the subresultant pseudo-remainder sequences ( https://en.wikipedia.org/wiki/Polynomial_greatest_common_divisor#Pseudo-remainder_sequences ) as described in Wikipedia. However, when ...
0 votes
1 answer
110 views

How to arrange the polynomial according to the descending power of x?

The polynomial is as follows: xpoly=-a^2 b^2 + (b^2 + a^2 k^2) x^2 + a^2 k^2 x0^2 - 2 a^2 k x0 y0 + a^2 y0^2 + x (-2 a^2 k^2 x0 + 2 a^2 k y0) How to get the ...
1 vote
1 answer
53 views

How to convert a list of polynomials with different degrees into a list of lists?

I asked a question about converting a polynomial into a list. Now I have a list of polynomials. For example, ...
1 vote
0 answers
37 views

Performing difference in extracting coefficients from a sum

I have an expression ...
1 vote
2 answers
458 views

Polynomial PowerMod

Is there equivalent of PowerMod for polynomials in Mathematica? We have Mod[a^e,m]==PowerMod[a,e,m], $a$, $e$ and $m$ all ...
4 votes
3 answers
378 views

Efficiently strip off coefficients in front of variables?

I am working with multivariate polynomials and need a very efficient way to decompose monomials into coefficients and pure monomials. for instance consider variables ...
2 votes
1 answer
74 views

How to write a^2 - a b + b^2 - a c - b c + c^2 has the form a^2 + b^2 + c^2 - a b - b c - c a?

I tried ...
1 vote
1 answer
43 views

Solve polynomial functions with four unknown variables [duplicate]

I want to make curve with polynominal function like this figure but only 2 known point to solve this one. I try to used solve function for this one. but the result is not the same like the figure. ...
0 votes
0 answers
26 views

How to Factor Out Polynomial Terms which Have Same Coefficient Differing Only by Sign

I'm new to the Mathematica StackExchange. I am trying to simplify a polynomial in Mathematica. In the following, I only show a part of the polynomial Here I want to combine terms with the same ...
1 vote
1 answer
62 views

Finding constant term in product expression

I've an expression which is product of 20 or more factors of polynomial, something like $$\left(1-\frac{pq}{z^i}\right)(1+pq z^j+z^k)$$ and I want to find coefficient of $z^0$. SeriesCoefficient works ...
1 vote
1 answer
338 views

Formatting results of a polynomial long division (Extension for finite fields)

I've just read Formatting results of a polynomial long division with great interest. For my teaching purposes it would be great to enable the code given in that post to treat polynomial division with ...
5 votes
2 answers
128 views

Simplifying a logical expression with multivariate polynomials

I have several expressions similar to a (a b + c d) != 0 || b (a b + c d) != 0 || c (a b + c d) != 0 || d (a b + c d) != 0 and I would like to reduce them ...
0 votes
2 answers
63 views

A polynomial with positive integer powers is being displayed as a transcedental function (LerchPhi)

I am defining a polynomial as follows: ...
0 votes
0 answers
81 views

How can I approximate a function with composite polynomials?

I know that to approximate a function with for example $f(x) = \sin(x)$ using polynomials with degrees up to 4, I can use the Fit function : ...
6 votes
3 answers
710 views

Implementation of a complex recurrence relation for polynomials

I would like to implement the recurrence relation for the polynomials $U_n(x)$ that appear in the large-order asymptotics of the modified Bessel functions. The recurrence in question is $$U_{n+1}(x)=\...
0 votes
1 answer
89 views

Finding the solution for the cubic formula over NonNegativeReals

The standard cubic polynomial is: $ax^3+bx^2+cx + d$. And when I used my function: ...
1 vote
1 answer
157 views

How to factor a quartic equation whose coefficient has unknown parameters?

i'm trying to see if a quartic equation I obtained can be factored into simpler forms, such as the product of two quadratics. The problem is that their coefficients are some complex expressions in ...
0 votes
1 answer
46 views

How to factorize high-order polynomials that have only complex roots?

I have polynomials like this: ...
2 votes
1 answer
194 views

How to verify a solution of an ordinary differential equation?

Given an ordinary differential equation with initial conditions eq = u a[u] + (16 + u^2 + 2 u a[u] (12 + u a[u] (6 + u a[u]))) a'[u] == 0 ic = a[0] == -1/2 How can ...
5 votes
3 answers
178 views

Calculating the basis set of quotient spaces

Having a polynomial $f(x,y)$, I would like to compute the following quantity \begin{equation*} {\mathbb C}[X,Y,Z]/\langle f_{x}, f_{y}, f_{z} \rangle, \end{equation*} where $f_{x},f_{y},f_{z}$ are, ...
12 votes
5 answers
8k 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 1....
1 vote
2 answers
139 views

Elegant way to restrict PolynomialMod to non-negatives

Update at the bottom! PolynomialMod[4 + 10x, 1 + 2x] returns -1. Instead, I'd like to get 4, ...
1 vote
1 answer
71 views

Expanding polynomials using valuation

I would like to expand the polynomial $p(\lambda) = \sum_{i=0}^{d} a_{i} \lambda^{i}$, as $F(p(\lambda), \lambda_{0})= \min_{j} [ val(a_{j}) + j \lambda_{0} ] $ with $\lambda_{0}$ being a real ...
2 votes
3 answers
490 views

How to generate a set of orthogonal polynomials with a special weight function?

Regarding to $e^{-ax-bx^2}$ as the weight function , how can I generate a set of orthogonal polynomials? I just know the Orthogonalize command in Mathematica which ...
1 vote
1 answer
95 views

Rearranging a simple algebraic expression

I have a polynomial of variables $x,y$, where $|x|<1$ and $|y|<1$. When I apply the Simplify function to this expression, I get an expression of the form $(x-...
0 votes
0 answers
54 views

Solving polynomial roots of any degree only by Vieta's formula

I have already wrote the code that solves it for quadratic formula and I'm curious if this is possible to make that function work with any kind of polynomial(higher degrees) and solving roots only ...
1 vote
2 answers
120 views

How to extract coefficients of polynomial formatted like this?

I want to extract coefficients. ...
16 votes
2 answers
666 views

Does NRoots own an abstract counterpart? If not, can we write one?

We know when solving linear algebra equations, despite its abstract syntax, LinearSolve is much faster compared to Solve: ...
4 votes
1 answer
316 views

How to make a polynomial so that f(i) = 1/(2^i)?

I know that, sequence has formula $f(n) = \dfrac{1}{2^n}$ satifying the conditions $f(1)=\dfrac{1}{2}$, $f(2)=\dfrac{1}{4}$, $f(3)=\dfrac{1}{8}$, $f(4)=\dfrac{1}{16}$. Now I am trying to find a ...
0 votes
1 answer
41 views

How to make MMA distinguish between symbolic coefficients and variables when doing factorization? [closed]

I what to factor a polynomial with complicated symbolic coefficients Factor[p0^2 + k^2 r^2 \[Tau]^2 - 2 k p0 r \[Tau]^2 \[Omega] + p0^2 \[Tau]^2 \[Omega]^2] In ...
0 votes
1 answer
96 views

Compute all bivariate polynomials over GF(2) of degree d or less and evaluate them at certain polynomial input

I want to compute all $<=d$ degree bivariate polynomials of form $f_1(x)g_1(y) + f_2(x)g_2(y)$, over field $GF(2)$, and evaluate them at a certain polynomial input for eg $d = 1$, evaluation at $(p^...
3 votes
1 answer
151 views

Cannot solve this polynomial equation

I'm trying to solve the following polynomial equation for $x$: $$ (qx)^\alpha (1-qx)^{1-\alpha} = [(1-q)x]^\beta [1-(1-q)x]^{1-\beta} $$ where $\alpha, \beta, q, x$ are all strictly between 0 and 1. ...
2 votes
7 answers
329 views

How can I get a list of symmetric polynomials?

Q1: Suppose I have an expression like this one: $(1+x+y+z)^3$ How can I transform it into the following expression: $$\{1,x+y+z,x^2+y^2+z^2,xy+xz+yz,x^3+y^3+z^3,x^2y+x^2z+xy^2+y^2z+xz^2+yz^2,xyz\}$$ ...

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