Questions tagged [elimination-theory]
Elimination theory is the study of necessary and sufficient conditions for polynomial equations (E) to have solutions.In the homogeneous case, if the number of variables is equal to the number of equations, this leads to the study of the Resultant (polynomial in the coefficients of (E), obtained by "eliminating" the variables ). In the general case, one get a Resultant ideal, generated by polynomial relations in the coefficients of the equations (E).
20
questions
6
votes
2answers
599 views
How did Hilbert prove the Nullstellensatz?
All of the many proofs of the Nullstellensatz I have seen use results from long after Hilbert’s time: Zariski’s lemma, Noether normalization, the Rabinowitch trick, model theory, etc. How did Hilbert’...
5
votes
1answer
254 views
Abel-Ruffini theorem for systems of polynomial equations
I know the Abel-Ruffini theorem, which states that a general polynomial equation in one variable with degree $\geq 5$ is not solvable in radicals. I wonder whether there is a similar result for ...
4
votes
0answers
67 views
Division of bivariate polynomials
The following theorem (lemma 4.2.18 on page 97) is proven in thesis "Computationally efficient Error-Correcting Codes and Holographic Proofs" by Daniel Alan Spielman:
Let $E(X, Y)$ be a polynomial ...
0
votes
0answers
79 views
Does positivstellensatz and SOS proof system help here?
I have a system of $m$ homogeneous degree $2$ polynomial equations in $\mathbb Z[x_1,\dots,x_n]$ where $m=poly(n)$. Take
$$f_1(x_1,\dots,x_n)=0$$
$$\dots$$
$$f_m(x_1,\dots,x_n)=0$$
to be the system.
...
0
votes
0answers
109 views
Do many homogeneous polynomials help in faster integer root extraction?
Given $n$ homogeneous algebraically independent total degree $2$ polynomials with no $x_1^2,\dots,x_n^2$ variable in $\mathbb Z[x_1,\dots,x_n]$ with promise that it has non-zero integer roots with ...
0
votes
0answers
119 views
Final step in Coppersmith?
In the final step in Coppersmith technique we have $n$ polynomials (possibly non-homogeneous) in $\mathbb Z[x_1,\dots,x_m]$ where $m\leq n$ and using elimination theory we extract the common integer ...
2
votes
0answers
152 views
Solving solutions to systems of polynomial equations over $\mathbb Z$
Macaulay Resultants help identify common root in $\mathbb P^{n-1}(\mathbb K)$ of $n$ homogeneous polynomials in $n$ variables when $\mathbb K$ is algebraically closed. Is it possible that some type of ...
18
votes
1answer
592 views
Counting real zeros of a polynomial
I recently came across a criteria to count the number of real zeros of a polynomial $P(x)$ with real coefficients. Unfortunately I cannot find the reference! The criteria is the following: Form the ...
3
votes
1answer
468 views
Question on the integral $\int_{-\infty}^{\infty} e^{a x^4+b x^3+c x^2+d x+f}\,dx$
From the wikipedia page on Gaussian integral https://en.wikipedia.org/wiki/Gaussian_integral the following formula holds:
$$\int_{-\infty}^{\infty} e^{a x^4+b x^3+c x^2+d x+f}\,dx =\frac12 e^f \ \...
5
votes
2answers
270 views
Discriminant of a composition of binary forms
Let $F(x,y), A(x,y), B(x,y)$ be homogeneous polynomials with integer coefficients (a.k.a. binary forms) such that the degrees of $A(x,y)$ and $B(x,y)$ match. Define
$$R(x,y) := F\left(A(x,y), B(x,y)\...
0
votes
0answers
124 views
Computer algebra programs for dummies
In the way of my investigations I have encounter the following computational problem: I have a system of 5 algebraic equations and I want to eliminate 4 of them. I also need to do a functional ...
1
vote
0answers
40 views
Elimination theory for variables packaged in a matrix
I am wondering if the elimination theory in computational algebraic geometry can be more efficiently carried out if all variables lies within some given matrices.
For instance, consider the following:
...
2
votes
0answers
138 views
Algebraic approach to showing trigonometric equations have no solution
I have very little background in algebra and algebraic geometry, so please bear with me.
I am trying to show that certain systems of trigonometric polynomial equations generally have no solution. One ...
12
votes
0answers
237 views
Entropy in elimination theory, or a brief remark by Gelfand-Kapranov-Zelevinsky
In the introduction to their book "Discriminants, resultants and multidimensional determinant", the authors state a very intriguing observation concerning the coefficients of monomials appearing in $A$...
10
votes
1answer
2k views
Why A. Weil considered elimination theory to be eliminated?
It is well known that André Weil declared, in the 1940's, that elimination theory must be eliminated from algebraic geometry. I would like to understand his mathematical reasons to adopt such an ...
-2
votes
1answer
452 views
Why is any maximal minor of the Bezoutian matrix divisible by the resultant?
I'm referring to Emiris and Mourrain's paper "Matrices in Elimination Theory," Theorem 3.13. Toward the end of the proof, it says that, just because $(f_1,\ldots,f_{n+1})$ is dense in ${\cal Z}({\rm ...
1
vote
1answer
445 views
Calculating the images of varieties under projections
Dear all,
I am interested in the following basic and fundamental question in elimination theory: given a variety in some product space $Z\subseteq X\times Y$, how could I explicitly calculate the ...
1
vote
5answers
350 views
Interpolating for particular coefficients
Say $F(X) \in \mathbb{Z}[X]$ is an even degree polynomial of degree $2n$.
One needs to evaluate $F(X)$ at $O(n)$ points to interpolate and get all the coefficients of $F(X)$.
However say I need ...
2
votes
3answers
1k views
General hyperplane sections and projection from a point
Let $k$ be an algebraically closed field, and consider some subscheme $X\subset \mathbb{P}_k^n$. Let $x$ be a closed point of $X$, and $H$ a general hyperplane containing $x$. There is a regular map $\...
5
votes
1answer
2k views
Polynomial with two repeated roots
I have a polynomial of degree 4 $f(t) \in \mathbb{C}[t]$, and I'd like to know when it has two repeated roots, in terms of its coefficients.
Phrased otherwise I'd like to find the equations of the ...
