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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Degree bounds and coefficient size in elimination theory?
Suppose we have polynomials is of form
$$h_1(x_1,\dots,x_{dn})-c_1\in\mathbb Z[x_1,\dots,x_{dn}]$$
$$\vdots$$
$$h_{nd}(x_1,\dots,x_{dn})-c_{nd}\in\mathbb Z[x_1,\dots,x_{dn}]$$
where $h_1(x_1,\dots,x_{...
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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 ...
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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 ...
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1answer
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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 ...
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1answer
301 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:
$$\int_{-\infty}^{\infty} e^{a x^4+b x^3+c x^2+d x+f}\,dx =\frac12 e^f \ \...
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2answers
239 views
Discriminant of a composition of binary forms
Let $F(x,y), A(x,y), B(x,y)$ be homogeneous polynomials with integer coefficients (a.k.a. binary forms) such that the degrees of $A(x,y)$ and $B(x,y)$ match. Define
$$R(x,y) := F\left(A(x,y), B(x,y)\...
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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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0answers
133 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 ...
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223 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 $A$...
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1answer
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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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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 ...
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1answer
344 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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5answers
327 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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3answers
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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 $\...
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1answer
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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 ...
