All Questions
10 questions
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Primitive element theorem for algebraic functions
Given a function $f(x) : \mathbb{R}^n \to \mathbb{R}$, we call it algebraic if it satisfies a polynomial equality $g(y, x) = 0$.
This is analogous to an algebraic number being the root of a univariate ...
- 869
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73
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Relations between non-negativity of multivariate polynomials and SOS over gradient ideal
We know there is a necessary condition for the non-negativity of multivariate polynomials in the paper "Sum of Squares Decompositions of Polynomials over their Gradient Ideals with Rational ...
- 12.9k
1
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280
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Algebraic independence criterion
Is there any criterion for checking algebraic independence of a set of polynomials in $n$ variables in terms of the leading monomials with respect to some monomial order ? The Jacobian criterion is ...
- 13.9k
1
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Finding multivariate binomials with a common zero [closed]
I have a problem for which I have to find binomials over a multivariate polynomial Ring which all have a common zero.
Let $\mathbb{F}[x_1,\dots,x_n]$ be some multivariate polynomial ring over some ...
1
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1
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174
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Sufficient syntactic conditions for zero-dimensionality of polynomial systems
Consider a system $S$ of polynomial equations, $p_1=0,...,p_m=0$, for $p_i\in K[x_1,...,x_n]$, for a field $K$: the system $S$ is zero-dimensional if it has finitely many solutions. It is well-known ...
- 33.8k
1
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139
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Polynomial systems and algebraic functions
An algebraic function $y(x)$ is defined as the solution of a polynomial equation of the form $p(x,y)=0$, that is one making the identity $p(x,y(x))=0$ true --- in either analytical or formal power ...
- 333
4
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206
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Software computing dimension and degree
Assume a projective scheme $X_{k_1,\dots,k_r}\subset\mathbb{P}^n$ is given as the set of common solutions of homogeneous polynomials $F_1(x_0,\dots,x_n),\dots,F_s(x_0,\dots,x_n)$, where the $F_i$ ...
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590
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A question on a Macaulay2 computation
I have an ideal $I$ generated by quadratic and cubic homogeneous polynomials in $10$ variables.
Macaulay2 tells me that $I$ defines an irreducible variety $X$ of dimension $5$ and degree $10$ in $\...
- 3,093
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273
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Solving polynomial inequalities -- efficient Positivstellensatz on a computer
I have about twenty five (multilinear) polynomials $f_1(\mathbf{x}), f_2(\mathbf{x}), \dots, f_{25}(\mathbf{x})$ all in fifteen variables and I would like to decide if there is a $\mathbf{y} \in [0,1]^...
- 54.2k
1
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146
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Omitting constraints of polynomial system
Let $n_1, n_2 \geq 1$ be known integer constants.
Suppose that we have the following system of $n$ polynomial inequalities
for which we know that there exists a feasible solution $(p_1, p_2) \in (0,1)...
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