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).

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2 votes
1 answer
86 views

Library/Database of parametric polynomial systems

Could anyone please recommend a known website where I can find a database/library that has systems of polynomial equations with $n$ variables and $m$ parameters? I need some real examples to test my ...
5 votes
1 answer
305 views

Does the main theorem of elimination theory with $\mathbb{Z}$-coefficients imply that projective varieties are complete?

I have a question concerning the completeness of projective varieties. Let $k$ be an algebraically closed field. By the "main theorem of elimination theory" I mean the following result: Let $...
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2 votes
3 answers
436 views

Useful software for variable elimination

I have three non-homogeneous trivariate polynomials in $\mathbb Z[x,y,z]$ and I want to eliminate the variables $y$ and $z$ to get a polynomial in $x$. The monomials of the polynomials are $\{1,x^4,x^...
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5 votes
1 answer
274 views

Multiple root of resultant

Let us suppose that we have two polynomials $F_1(x,y)$ and $F_2(x,y)$. Generally speaking, each of them defines a curve on the plane and the system of polynomial equations defined by them computes the ...
4 votes
2 answers
294 views

$n-1$ quadratic forms for $n$ variables

If we have $n-1$ quadratic forms for $n$ variables $x_i$, $$p_i(x) = M^{(i)}_{jk} x_j x_k$$ for $1\leq i \leq n-1$ and $1 \leq j,k \leq n$ then the zeros of all $p_i(x)$, $$p_i(x) = 0$$ is generically ...
2 votes
0 answers
119 views

Degree of polynomials describing projection of algebraic set

Consider an algebraic subset $V\subseteq \mathbb{R}^{n+1}$ defined as the zero set of polynomials ${f_i}$ and the projection map $\pi: \mathbb{R}^{n+1}\to \mathbb{R}^n$ deleting the last entry. By the ...
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1 vote
0 answers
87 views

Expression for the single common root

Let $ \mathbb{F} $ be a field, consider the polynomial ring $ \mathbb{F} \left[ x\right] $ and suppose that the polynomials $f,g \in \mathbb{F} \left[ x\right]$ have degrees $2,2^n$, respectively, ...
4 votes
0 answers
195 views

Effective bounds for a Bertini-type result

Suppose $X$ is a projective subvariety of $\mathbb{P}^n$ of codimension $r$ over $\mathbb{C}$, defined set-theoretically by $r$ homogeneous polynomials $P_1,\dots,P_r$ of degree at most $d$. By ...
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11 votes
1 answer
876 views

History of Sylvester's resultant?

Suppose that we have two polynomials that split: $$\begin{align*} f(x)=\sum_{k=0}^d a_{d-k}x^k&=\prod_{i=1}^d (x-\lambda_i),\\ g(x)=\sum_{k=0}^e b_{e-k}x^k&=\prod_{j=1}^e (x-\mu_j).\\ \end{...
0 votes
1 answer
102 views

On a sum of squares representation

We know $p a^2+q b^2+r ab$ can be represented as square (trivially) when $$p,q\geq0$$ $$r^2=|4pq|$$ holds and as a sum of squares (again trivially) of form $(m a+n b)^2$ under readily explainable ...
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7 votes
2 answers
884 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’...
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5 votes
1 answer
370 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
0 answers
87 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
0 answers
115 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. ...
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0 votes
0 answers
119 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 ...
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0 votes
0 answers
124 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 ...
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2 votes
0 answers
162 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 ...
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19 votes
1 answer
645 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
1 answer
576 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 \ \...
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5 votes
2 answers
345 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)\...
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0 votes
0 answers
139 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 ...
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1 vote
0 answers
45 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: ...
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2 votes
0 answers
145 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
0 answers
267 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 $A$...
11 votes
1 answer
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 ...
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-2 votes
1 answer
480 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
1 answer
574 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 ...
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1 vote
5 answers
419 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 ...
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2 votes
3 answers
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
1 answer
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 ...