factorization – Can Mathematica factor multivariate polynomials with 4 or more variables? And with high degrees (>10)

The Mathematica documentation clearly states

the Wolfram Language routinely factors degree-100 polynomials in 3 variables

I’m interested in factoring systems of polynomials in as many as 10 or 20 variables. The systems I have are sparse in the sense that if there are 20 variables then likely no more than 3-6 variables will appear per equation and there will likely only be two terms per equation. I’ve used Solve to test on some small systems with success. To be clear, we don’t actually need to factor the polynomials necessarily. If we factor them, then we have what we need. What we really want are the roots to the system with respect to the symbolic coefficients. Solve worked for some small test systems. Factor would also work for a single polynomial, but we need a system of polynomials and I don’t see that Factor will take a system.

I’ve looked at papers by searching scholar.google.com and it seems our problem is a solved problem in mathematics. Algorithms seem to exist for such a problem, but I’m even unclear on this since the papers are too densely packed with math for me to easily understand.

Any help would be appreciated.

linear algebra – Orthonormal polynomials and recurrence relation

I read a book where there is this example:

Given orthonormal polynomials $pi_1,pi_2,dots$ then the recurrence relation holds

$$xpi_n(x)=a_npi_{n+1}(x)+b_npi_n(x)+a_{n-1}pi_{n-1}(x)$$

with $a_n=frac{gamma_n}{gamma_{n+1}}$ where $pi_n(x)=gamma_nx^n+dots$.

I have especially troubles to understand why $a_n=frac{gamma_n}{gamma_{n+1}}$ and there is no derivation.

I understand this part:

If we define monic polynomials $p_n(x)=frac{pi(x)}{gamma_n}$ we can write $p_n(x)$ as the sum of 3 terms
$$x p_n(x)=sumlimits_{k=0}^{n+1}p_k(x)frac{langle xp_n,p_krangle}{langle p_k,p_krangle}\
=frac{langle x p_{n},p_{n+1}rangle}{langle p_{n+1},p_{n+1}rangle}p_{n+1}(x)+frac{langle xp_{n},p_nrangle}{langle p_{n},p_{n}rangle}p_{n}(x)+frac{langle p_n,x p_{n-1}rangle}{langle p_{n-1},p_{n-1}rangle}p_{n-1}(x)$$

Hence we get a recurrence for $p_n$.
$$xp_n(x)=a_np_{n+1}(x)+b_np_n(x)+c_{n}p_{n-1}(x)$$

But here I am lost. If we replace the $p_n$ with $pi(x)/gamma_n$ I get
$$xpi_n(x)=a_nfrac{gamma_n}{gamma_{n+1}}pi_{n+1}(x)+b_npi_{n}(x)+c_{n-1}frac{gamma_n}{gamma_{n-1}}pi_{n-1}(x)$$
and $$a_n=frac{langle x p_{n},p_{n+1}rangle}{langle p_{n+1},p_{n+1}rangle}=gamma_{n+1}^2langle x p_{n},p_{n+1}rangle$$

polynomials – If $a^{1/a}=b^{1/b}=c^{1/c}$ and $a^{bc}+b^{ac}+c^{ab}=729$, find the value of $a^{1/a}$

If $a^{1/a}=b^{1/b}=c^{1/c}$ and $a^{bc}+b^{ac}+c^{ab}=729$, which of the following equals to $a^{1/a}$?

  1. $sqrt(abc){81}$
  2. $sqrt{2}$
  3. $sqrt(abc){27}$
  4. $sqrt(abc){9}$

This question is from the book, Mathematics, Class 9 (The IIT Foundation Series) , page number 1.25, question number 58.

My attempts to solve this question have failed several times. However, I did find the value of $a^{1/a}$ but not in the correct format. Below is my method to do so.

$$a^{1/a}=b^{1/b}=c^{1/c}$$
$$Rightarrow sqrt(a){a}=sqrt(b){b}=sqrt(c){c}$$
$$Rightarrow (sqrt(a){a})^{abc}=(sqrt(b){b})^{abc}=(sqrt(c){c})^{abc}$$
$$Rightarrow a^{bc}=b^{ac}=c^{ab}$$

Here I conclude our first equation, $a^{bc}=b^{ac}=c^{ab}$. Moving on to the next equation, we have:

$$a^{bc}+b^{ac}+c^{ab}=729$$
$$Rightarrow a^{bc}+b^{ac}+c^{ab}=729$$
$$Rightarrow a^{bc}+a^{bc}+a^{bc}=729$$
$$Rightarrow 3a^{bc}=729$$
$$Rightarrow a^{bc}=243$$
$$Rightarrow (a^{bc})^{1/abc}=243^{1/abc}$$
$$Rightarrow a^{1/a}=sqrt(abc){243}$$
$$Rightarrow a^{1/a}=sqrt(abc){3^5}$$

Here I finally find the vale of $a^{1/a}$ as $sqrt(abc){3^5}$. However, none of the options match with my result. Please help me to solve the question completely. Thanks!

Nowhere negative polynomials form a semialgebraic set

Let $P_{d, n}$ be the space of polynomial maps $mathbb{R}^nto mathbb{R}$ of degree at most $d$.

Is the subset $Ssubset P_{d, n}$ of nowhere negative polynomials semialgebraic?

Find a pair $p,q$ of polynomials for which $p$ $circ$ $q$ = $q$ $circ$ $p$.

Can anybody tell me what does the notation $circ$ mean in the below question ?$\$
Find a pair $p,q$ of polynomials for which
$p$ $circ$ $q$ = $q$ $circ$ $p$.

real analysis – Variation of the sum of absolute values of coefficients for shifted Chebyshev polynomials

Let $rho in )0,1($, $varepsilonin(0,rho)$, $k in mathbb{N}^*$ and
$$P^varepsilon_k(X) = tfrac{T_kleft(tfrac{2(X+varepsilon)}{rho+varepsilon}-1 right)}{left|T_kleft(tfrac{2(1+varepsilon)}{rho+varepsilon}-1 right) right|}$$
where $T_k$ is the Chebyshev polynomial of first kind with degree $k$, i.e. $T_k(cos(theta)) = cos(ktheta),; forall theta in mathbb{R}$.

I would like to show that the function $varepsilon rightarrow | P^varepsilon_k |_1$ (with $|cdot|_1$ is the sum of the absolute value of the polynomial’s coefficients) is decreasing on $(0,rho)$. It is true for $varepsilon$ near $0$ and seems true numerically for all $varepsilonin(0,+infty($.

Case $varepsilon$ close to 0

We have from the definition of $T_k$ that its roots are the $(z_i)_{iin(0 ,k-1)} = (cosleft(tfrac{2i+1}{2k}pi right))_{iin(0 ,k-1)} in (-1,1)$. The roots of $P^varepsilon_k(X)$ are the $z^varepsilon_i$ defined such that $tfrac{2z^varepsilon_i+2varepsilon}{rho+varepsilon} -1 = z_i $. This corresponds to
$$ z^varepsilon_i = tfrac{(rho+varepsilon)z_i + rho – varepsilon}{2} in (-varepsilon,rho), quad i=0,dots,k-1.$$
The smallest root $z^varepsilon_{k-1} = tfrac{(rho+varepsilon)cos(tfrac{2k-1}{2k}pi) + rho – varepsilon}{2}$ is nonnegative for $varepsilonin(0,rhotfrac{1+cos(tfrac{2k-1}{2k}pi)}{1-cos(tfrac{2k-1}{2k}pi)})$. This means that this for choice of $varepsilon$, all the roots of $P^varepsilon_k$ are nonnegative and therefore coefficients of $P^varepsilon_k$ alternate signs (we see that by expressing the coefficients using the roots). This fact implies that one can write $| P^varepsilon_k|_1 = left | P^varepsilon_k(-1) right|$ which can be expressed as
begin{align*}
left | P^varepsilon_k(-1) right| &= tfrac{left|T_kleft(tfrac{2(-1+varepsilon)}{rho+varepsilon} -1right)right|}{T_kleft(tfrac{2(1+varepsilon)}{rho+varepsilon} -1right)}\
& = tfrac{left(1 +tfrac{rho-varepsilon}{2} -sqrt{(1+rho)(1-varepsilon)}right)^k +left(1 +tfrac{rho-varepsilon}{2} +sqrt{(1+rho)(1-varepsilon)}right)^k}{left(1 +tfrac{rho-varepsilon}{2} -sqrt{(1-rho)(1+varepsilon)}right)^k +left(1 +tfrac{rho-varepsilon}{2} +sqrt{(1-rho)(1+varepsilon)}right)^k},
end{align*}

where we use the formula $T_k(x) = tfrac{1}{2}left(left(x – sqrt{x^2-1} right)^k+left(x + sqrt{x^2-1} right)^k right)$ for $|x| geq 1$. Finally, a simple study of function shows that the numerator is decreasing and the denominator increasing.

Additional info

Note that $P^varepsilon_k$ is solution of the optimization problem
$$ underset{substack{pinmathbf{R}_k(X)\p(1)=1}}{min}underset{xin(-varepsilon,rho)}{max}|p(x)|. $$

Any ideas or pointers are welcome, thanks for your help.

equation solving – Decomposing polynomial as a sum of polynomials multiples of two polynomials

Suppose that I have a polynomial $f(s,t)$ with coefficients in $R=mathbb{R}(x_{1},dots, x_{n})$ and in the variables $s,t$ and I know that $f(s,t)$ can be expressed as $(s-t)f(s,t)=p(s,t)h(s)+q(s,t)h(t)$ with $h$ having coefficients also in R (so these coeffcients are actually polynomials) but I do not know who are $p,qin R(s,t)$. How can I make Mathematica solve the equation $(s-t)f(s,t)=p(s,t)h(s)+q(s,t)h(t)$ finding the polynomials $p,q$ in the variables $s,t$ with polynomials coefficients in $R$? Thanks!

nt.number theory – Integer valued polynomials and polynomials with integer coefficients

It is well known that the subring $S$ of integer valued polynomials ${mathbb Q}(x)$ is generated by the binomial functions $P_n={x choose n}$. One can ask a dual question: how to characterize the polynomial functions ${mathbb Z} to {mathbb Z}$ which come from an element in ${mathbb Z}(x)$.
I understand one can write down derivative as a (terminating) series of difference derivatives and thus express each coefficients in terms of values of the polynomial but does this (or another) procedure lead to a neat answer?

There is a necessary condition for a polynomial function $f:{mathbb Z} to {mathbb Z}$ to come from an element in ${mathbb Z}(x)$, namely for every $n$ the residue of $f(x) mod n$ depends only on the residue of $x mod n$. This necessary condition is not sufficient but am interested in the subring of elements in $S$
satisfying that necessary condition. Is there a nice set of generators and/or a basis?

inequalities – Does there exist a type of discriminant not only for irreducible polynomials but also for exponential functions, logarithm functions ? Thanks a rl lot

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equation solving – Expanding function of polynomials

I currently have a function $f(x) = beta_0 + beta_{1}(x-bar{x}) + beta_{2}(y-bar{y}) + beta_{3}(x-bar{x})^2 + dotsc + beta_{j}(y-bar{y})^5$ and am trying to develop this in Mathematica. Is it possible to provide a vector of these numbers to return a symbolic equation to expand the polynomials and give the result in terms of each variable? I imagine it would start as

F1(x_, xb_, y_, yb_) := ...

but am not sure where the betas come into play or how the expansion would occur.

Thanks in advance.