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Results tagged with polynomials
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user 32332
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 $m(t),n(t)$. …
- 12.1k
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. …
- 12.1k
2
votes
Irreducibility of trinomials over number fields
Schinzel: "Solved and unsolved problems on polynomials", $1995$. …
- 12.1k
7
votes
Accepted
Knots indistinguishable by HOMFLY
.
$$
This shows that the polynomials are different (except for obvious symmetries). …
- 12.1k
3
votes
Existence of non-trivial solution to non linear polynomial system
Using Groebner bases (or even direct computing) we see, that the solutions of your system $f_5=f_6=0$ are as follows:
Case 1: $a_1b_2-a_2b_1=0$. Then it follows $y^2+z^2=0$.
Over the real numbers th …
- 12.1k
2
votes
Accepted
"Symmetric" Polynomial 4-cocycles
Heaton showed that if the field $K$ has characteristic zero, then all polynomial cocycles are coboundaries. In characteristic $p$ this is not always the case, but the following result holds (which you …
- 12.1k
2
votes
"Degree 3 fields"
Such fields are related to real-closed fields, which satisfy an even stronger property. A field $K$ is called real-closed, if it is formally real, i.e., $-1$ is not a sum of squares in $K$, and no pro …
- 12.1k
8
votes
0
answers
397
views
When does the Lloyd polynomial have only integral roots?
For a $t$-error correcting code of length $n$ over the finite field $\mathbb{F}_q$, the Lloyd polynomial is given by
$$
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 …
- 12.1k
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 …
- 12.1k
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. …
- 12.1k