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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Resultants and elimination theory

Consider an ideal $I = \langle f_1,\dotsc,f_n\rangle$ in the ring $k[x_1,\dotsc,x_m]$. Define the $i$-th elimination ideal to be $I_i = I \cap k[x_{i+1},\dotsc,x_m]$. For any two polynomials $f$ and $...
giulio bullsaver's user avatar
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1 answer
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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 ...
Ayoola Jinadu's user avatar
5 votes
1 answer
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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 $...
Luvath's user avatar
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3 answers
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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^...
Turbo's user avatar
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5 votes
1 answer
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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 ...
MidEastGuest's user avatar
4 votes
2 answers
356 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 ...
Fetchinson0234's user avatar
2 votes
0 answers
157 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 ...
BGJ's user avatar
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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, ...
PalmTopTigerMO's user avatar
4 votes
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203 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 ...
Zeyu's user avatar
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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{...
Drew Armstrong's user avatar
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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 ...
VS.'s user avatar
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7 votes
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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’...
Fernando's user avatar
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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 ...
curiousStudent's user avatar
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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 ...
Maxim Nikitin's user avatar
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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. ...
VS.'s user avatar
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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 ...
VS.'s user avatar
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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 ...
Turbo's user avatar
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2 votes
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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 ...
Turbo's user avatar
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19 votes
1 answer
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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 ...
Michael Griffin's user avatar
3 votes
1 answer
616 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 \ \...
sam's user avatar
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5 votes
2 answers
372 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)\...
Anton's user avatar
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0 answers
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Computer algebra programs for dummies [duplicate]

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 ...
Johnny Cage's user avatar
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1 vote
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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: ...
Kevin Ye's user avatar
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2 votes
0 answers
147 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 ...
Yoni Rozenshein's user avatar
12 votes
0 answers
279 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 $A$...
Camilo Sarmiento's user avatar
12 votes
1 answer
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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 ...
Al-Amrani's user avatar
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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 ...
Zirui Wang's user avatar
1 vote
1 answer
648 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 ...
Zeyu's user avatar
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1 vote
5 answers
486 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 ...
Turbo's user avatar
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2 votes
3 answers
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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 $\...
Nathan Ilten's user avatar
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 ...
Andrea Ferretti's user avatar