Questions tagged [polynomials]

Questions on the functionality operating on polynomials

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Tropical Operation Conversions: Multiple Operations

I apologize if my syntax is bad. I looked around, but didn't see anything on this in the stack exchange. I would like to be able to transform a classical real-valued polynomial to a tropical ...
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1 vote
1 answer
88 views

While inside a function body does not work as expected [closed]

I'm trying to write a function that return a randomly created rational polynomial function that has a discontinuities points. I already have a function that generates random rational polynomial (...
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  • 69
1 vote
1 answer
60 views

How to perform the first and second derivatives of Nth polynomials?

Suppose  $P(x,y)=\displaystyle\sum^{N}_{i+j=0}\alpha_{ij}x^{i}y^{i} \equiv \alpha_{00}+\alpha_{10}x+\alpha_{01}y+a_{11}xy+\ldots+a_{0N}y^{N}$ is a multivariable polynomial of $N$th degree. I want to ...
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5 votes
1 answer
166 views
+50

Rational roots of x^5 + a x + b

I'm validating a formula from these article: https://www.annals-csis.org/Volume_20/drp/pdf/164.pdf https://people.math.carleton.ca/~williams/papers/pdf/185.pdf If irreducible rational equation $x^5 +...
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  • 3,644
0 votes
0 answers
57 views

How to speed up Resultant?

I am experimenting with the polynomials Y and T: i1 = 9; i2 = 4; Y = Sum[ x^i y[i], {i, 0, i1}]; T = Sum[ x^i t[i], {i, 0, i2}]; Timing[Resultant[Y, T, x]] Whith ...
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3 votes
2 answers
108 views

How to know the Galois Group of a polynomial is a solvable group?

ResourceFunction["StauduharGaloisGroup"][2 x^5+3 x^4+10 x^3+15 x^2+8 x+12,x]["GaloisGroup"] ...
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3 votes
1 answer
102 views

What is the fastest way to check whether a cubic equation is solvable in integers?

My code needs to check whether a lot of low-degree equations (usually quadratic and cubic) are solvable in integers. There are many equations, so the speed is crucial. Let us start with quadratic ...
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0 votes
0 answers
18 views

Why Cancel and PolynomialGCD treate option Extension in a different way?

I have difficulty understanding the following Cancel and PolynomialGCD examples. Expected: ...
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  • 2,847
4 votes
2 answers
135 views

Rewriting algebraic expressions as combinations of other expressions

I would like to write x^5 + 1/x^5 in terms of x^2 + 1/x^2, x^3 + 1/x^3 and ...
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  • 219
2 votes
4 answers
171 views

Constructing a specific polynomial Monomial ordering of $f(z,w)$ to convert to latex

Note: Afraid I did not initially adequately specify the precise Latex format of the algebraic expression. Edited to make more clear. I am having problems transferring polynomials in two variables $f(...
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  • 1,084
0 votes
1 answer
82 views

Finding the roots of a quartic [closed]

I try to find the roots of $f(x)=x^4+(4a-2)x^3+(6a^2-6a)x^2+(4a^3-7a^2)x+ a^4-2a^3 =0$ with $0<a<1$. How do i get with Mathematica all solutions (general expressions, real and complex)? With <...
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2 votes
0 answers
86 views

How to solve this equation analytically?

Why can't Mathematica 13 solve this equation in radicals? Solve[x^5 - 5*x^4 - 10*x^3 - 10*x^2 - 5*x - 1 == 0, x] // ToRadicals I also tried to use ...
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  • 71
2 votes
2 answers
222 views

Three equation solving analytically in Mathematica

Im doing coding in iterative schemes but I do not know to solve three equations analytically in Mathematica step by step. For example I given thre equations in the attached figure. I want to solve ...
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  • 169
0 votes
1 answer
94 views

Solving solvable quintics in trigonometric/radicals

If apply resultant to solvable quintic $f(x)$ as $Res_x(f(x),y+Rx+Sx^2+Tx^3)$ and associate output with trigonometric five-angle formula $\cos{5\theta} = 5cos{\theta} - 20cos^3{\theta} + 16cos^5{\...
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5 votes
2 answers
78 views

How to collect specified monomials in polynomial?

I'm not experienced in Wolfram Language, so couldn't you please help me with its syntax? Given the polynomial expression: ...
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8 votes
2 answers
117 views

Efficient algorithm for calculating polynomial that has roots of a certain form

I have looked into this specific question on Math.SE concerning a more "mechanical" approach to finding a polynomial $p \in \mathbb{Q}[x]$ satisfying $p(\sqrt{2}+\sqrt{3}) = 0$. The user MJD ...
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0 votes
0 answers
51 views

extremum points

I'm trying to find Min and Max points of the polynomial below, but it seems that the expression is very complicated to mathematical, I've tried a numerical function, such as NMinimize, NRoots, Solve, ...
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1 vote
0 answers
50 views

Can one collect with respect to two groups of variables?

Here is a toy example of what I want: given ...
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1 vote
0 answers
74 views

Mathematica doesn't simplify 1.` x

I'm debugging a program that uses polynomials with numerical coefficients and it turns out that mathematica does not simplify 1.` x[1]^2 x[2]^2 to ...
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  • 928
1 vote
0 answers
61 views

Easy upper and lower bounds for curve genus in Mathematica

I am writing a Mathematica code in which I need, at some point, to compute genus of a lot of curves given by polynomial equations in 2 variables in affine coordinates. Unfortunately, this is possible ...
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0 votes
0 answers
38 views

Why does Solve give different answers when specifying coefficients at different time?

Context I would like to find a specific - eventually complex - root of a 4th order polynomial as a function of a purely imaginary parameter $i\lambda$ who is continuous in $\lambda$. I am thus looking ...
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  • 185
1 vote
1 answer
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Sort polynomial by monomial degree [closed]

there is an equation : z = a^2 + a b + a a' + a a' a'' + a b' and i want to rearrange the equation by sorting the terms by number of variable multiplication within. If my English is not well, let me ...
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  • 61
1 vote
2 answers
133 views

Finding the definite integral of a square root of a polynomial function [closed]

I am trying to find the arc length of the function: f(x)=-0.005632x^7 + 0.08969x^6-0.5346x^5 + 1.364x^4 -0.8671x^3 -2.005x^2 + 3.038x + 0.4182 I am using this formula: My boundaries are 0 and 5. ...
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  • 11
1 vote
3 answers
104 views

What is the most efficient way to turn a metric formula into a metric tensor?

I have a metric formula: ds=(-dt^2)*(c3+a3*t)^2+(dxC^2*(t0^2+t1^2)^2)/(4*t0^4)+ (dxM^2*(t0^2+t1^2)^2)/(4*t0^4)+(dxY^2*(t0^2+t1^2)^2)/(4*t0^4) How do I turn this ...
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  • 1,481
1 vote
0 answers
42 views

Question about NSolve

there! For some purposes I need to be able to write a code in C# that could find all roots of the equations system, consisting of two-variable polynomial expressions, like the folliwing: ...
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1 vote
2 answers
95 views

why ContourPlot3D does not give plot?

I wanted to plot the intersections using ContourPlot3D in order to observe how do solutions look like, however, the following code does not work: ...
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11 votes
2 answers
626 views

Out of memory when computing a coefficient of a large symmetric polynomial

...
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2 votes
1 answer
70 views

Plotting of a polynomial function via two approaches

I am trying to plot a polynomial via two approaches: Take CoefficientList of the polynomial and then take Internal`FromCoefficientList, and finally replacing the value of the free parameter with a ...
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1 vote
1 answer
150 views

Homework from studies! - module computation of an approximating function

Polynomial approximation The aim of the task: to write a program in the form of a module in the Mathematica® machine code computation of an approximating function F (·) for an approximated ...
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1 vote
0 answers
73 views

How to solve this quintic polynomial with the solutions in its simplest final form?

Consider the following matrix: ...
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  • 465
0 votes
1 answer
41 views

Create a general polynomial function [duplicate]

I search for a simple procedure to create a polynomial function with unknown coefficients. For example if I asked for the most general polynomial of two variables 'x' and 'y', of second degree, it ...
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1 vote
1 answer
65 views

Plotting parts of a polynomial

Consider the following polynomial fun = 2 x + 34 x^2 - 5 y + 4 y^3; when I try fun[[1]] I get ...
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  • 1,047
6 votes
2 answers
191 views

Efficient way of finding polynomial relation

I have a list p of polynomials of k, and a list of "parameters" x,y,z which are ...
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  • 279
2 votes
1 answer
35 views

CoefficientRules numericizing parameters

I ran into a problem with CoefficientRules behaving differently with exact and numerical coefficients, where in the numerical case it behaves as if the entire ...
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  • 21
0 votes
2 answers
104 views

Problem about vanishing determinant of a matrix with NSolve

When I try to solve my question according to @DanielLichtblau's comments, I encountered another issue, with which I have been struggling for a whole night. Taking a matrix for example, ...
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  • 1,197
6 votes
2 answers
133 views

Reduce the number of equations

I have following 16 equations with 15 variables (XX is a variable, etc). I know there are only 6 independent variables. How can I reduce the number of equations below to get 9 independent equations? <...
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  • 657
0 votes
1 answer
47 views

Exclude terms in replacement rules application [closed]

Suppose I consider a polynomial of type A=B x y + C x and define a replacement rule rrule=(# /. x y -> 1) &; The result ...
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3 votes
0 answers
64 views

Is it intended that CoefficientRules permutes exponent vectors when given a monomial order?

I noticed an odd behavior in CoefficientRules when you give it a monomial order: ...
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1 vote
0 answers
26 views

N on equation with numbered variables [duplicate]

Say I have a system of polynomial equations in the variables x[1],...,x[n]. Sometimes want to make such equations numerical (as in: the coefficients become floats) ...
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  • 928
0 votes
1 answer
60 views

Minimal polynomial of an element in a polynomial quotient ring

Let $p(z)$ be a (not necessarily irreducible) polynomial in $\mathbb{Q}[z]$. Is there a built-in function I can use for determining the minimal polynomial of an element $\overline{q(z)} \in \mathbb{Q}[...
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  • 101
0 votes
0 answers
95 views

Algorithms or Mathematica definition for functions ideals of polynomial Rings

This question was well posed four years ago without answer at Operations on ideals of polynomial rings. I asked again in Nov 2021 and posted possible function defininitions. Lichtblau commented ...
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2 votes
2 answers
302 views

How can I find the only real and then the smallest root of a 4th-order polynomial?

I just want to solve the below polynomial for real roots only, where I have mentioned the conditions on all variables, a,d,m,L. ...
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1 vote
1 answer
65 views

Replace coefficients in a polynomial

I want to replace the coefficients of a polynomial by new coefficients. For example, my polynomial of variable 'r' is in the picture I want to replace the coefficients with 'a subscripts' similar to ...
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  • 11
0 votes
0 answers
45 views

Reduce returning redundant solutions

When I call reduce on the following system: ...
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7 votes
1 answer
216 views

Solving quadratic equations bug in Mathematica 12.3

There seems to be some sort of bug in mathematica, where sometimes it will treat a quadratic equation as a quartic equation. For example: ...
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2 votes
0 answers
66 views

Why is PolynomialQ[y^2,x] True? [closed]

Why is PolynomialQ[y^2,x] True? The documentation says PolynomialQ[] returns ...
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1 vote
0 answers
59 views

Expand, simplify, and get rid of Abs[] of a multi-variable polynomial [closed]

Given a polynomial that $poly = ((-1 + t_1^2) (-1 + t_2^2) (1 - t_2^2 + t_2^4))/(\sqrt{t_1^2} \cdot(t_2^2)^{3/2})$. I would like to expand and simplify this polynomial into the form $poly = \sum_{i\in ...
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2 votes
0 answers
72 views

Mathematica package for computing Macdonald polynomials

I want to implement computation of Macdonald polynomials in mathematica. A similar question was raised in another question 5 years ago (Macdonald-Koornwinder polynomials?), but received no clear ...
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  • 133
2 votes
0 answers
60 views

Extracting variables [closed]

Similar to this perhaps, but I prefer a built-in function. I have defined the expression L = a*b*c. Is there a Mathematica function that can identify and extract ...
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  • 21
2 votes
1 answer
93 views

Factoring a polynomial over a number field

Consider a polynomial x^3-x-1 and let $\alpha$, $\beta$, $\gamma$ be three zeros of the polynomial where $\alpha \in \mathbb{R}$. Since $\beta \not \in \mathbb{Q}[\alpha]$ and the degree of the ...
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