# 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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### 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 ...
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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 ...
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### $n-1$ 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 ...
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### 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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### 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, ...
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### 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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### 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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### 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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### 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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### 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 ...
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### 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 $A$...
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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 ...
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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 ${\cal Z}({\rm ... • 511 1 vote 1 answer 574 views ### Calculating the images of varieties under projections Dear all, I am interested in the following basic and fundamental question in elimination theory: given a variety in some product space$Z\subseteq X\times Y$, how could I explicitly calculate the ... • 527 1 vote 5 answers 419 views ### Interpolating for particular coefficients Say$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 ... • 13.1k 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$\...
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 ...