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Questions in which polynomials (single or several variables) play a key role. It is typically important that this tag is combined with other tags; polynomials appear in very different contexts. Please, use at least one of the top-level tags, such as nt.number-theory, co.combinatorics, ac.commutative-algebra, in addition to it. Also, note the more specific tags for some special types of polynomials, e.g., orthogonal-polynomials, symmetric-polynomials.

2 votes

Results for resolution of equations in polynomial ring

In general this is a difficult problem, but in special cases like $x^n+y^n=z^n$ in polynomials we can use Mason's theorem, which is about an analogue of the $abc$-conjecture for polynomials in $\mathbb … For the case $n=2$ we have the basic solutions $(a(t),b(t),c(t))=(m(t)^2-n(t)^2,2m(t)n(t),m(t)^2+n(t)^2)$ with polynomials $m(t),n(t)$. …
Dietrich Burde's user avatar
2 votes

"Degree 3 fields"

Such fields are related to real-closed fields, which satisfy an even stronger property. A field $K$ is called real-closed, if it is formally real, i.e., $-1$ is not a sum of squares in $K$, and no pro …
Dietrich Burde's user avatar
9 votes

Can you efficiently solve a system of quadratic multivariate polynomials?

It is not the fact that the polynomials are quadratic which helps, but rather other restrictions which will sometimes result in a more effective solution. …
Dietrich Burde's user avatar
7 votes
Accepted

Knots indistinguishable by HOMFLY

. $$ This shows that the polynomials are different (except for obvious symmetries). …
Dietrich Burde's user avatar
3 votes

Solving a System of Quadratic Equations

In answer to the comment, it is possible to compute a Groebner basis for the given example in a very short time, and from here one finds several solutions, e.g., a= 0, b= 0, c= (2260*h)/783, e= (8 …
Dietrich Burde's user avatar
2 votes

Irreducibility of trinomials over number fields

Schinzel: "Solved and unsolved problems on polynomials", $1995$. …
Dietrich Burde's user avatar
2 votes
Accepted

"Symmetric" Polynomial 4-cocycles

Heaton showed that if the field $K$ has characteristic zero, then all polynomial cocycles are coboundaries. In characteristic $p$ this is not always the case, but the following result holds (which you …
Dietrich Burde's user avatar
8 votes
0 answers
397 views

When does the Lloyd polynomial have only integral roots?

For a $t$-error correcting code of length $n$ over the finite field $\mathbb{F}_q$, the Lloyd polynomial is given by $$ L_t(n,x):=\sum_{j=0}^t(-1)^j\binom{x-1}{j}\binom{n-x}{t-j}(q-1)^{t-j}. $$ A well …
Dietrich Burde's user avatar
3 votes

Existence of non-trivial solution to non linear polynomial system

Using Groebner bases (or even direct computing) we see, that the solutions of your system $f_5=f_6=0$ are as follows: Case 1: $a_1b_2-a_2b_1=0$. Then it follows $y^2+z^2=0$. Over the real numbers th …
Dietrich Burde's user avatar
3 votes

Minimal representation of a polynomial as a linear combination of squares

of polynomials, i.e., $f=u^2-rv^2$ for some $r\in \mathbb{Q}$, and $u,v\in \mathbb{Q}[x]$. … As $-r$ is the sum of four rational squares, we can write $f$ as the sum of five squares of rational polynomials. …
Dietrich Burde's user avatar