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pro vyhledávání: '"Schweighofer, Markus"'
In 1995, Reznick showed an important variant of the obvious fact that any positive semidefinite (real) quadratic form is a sum of squares of linear forms: If a form (of arbitrary even degree) is positive definite then it becomes a sum of squares of f
Externí odkaz:
http://arxiv.org/abs/2310.12853
Autor:
Sawall, David, Schweighofer, Markus
With this article, we hope to launch the investigation of what we call the real zero amalgamation problem. Whenever a polynomial arises from another polynomial by substituting zero for some of its variables, we call the second polynomial an extension
Externí odkaz:
http://arxiv.org/abs/2305.07403
Autor:
Schweighofer, Markus
Chapters 1 to 4 are the lecture notes of my course "Real Algebraic Geometry I" from the winter term 2020/2021. Chapters 5 to 8 are the lecture notes of its continuation "Real Algebraic Geometry II" from the summer term 2021. Chapters 9 and 10 are the
Externí odkaz:
http://arxiv.org/abs/2205.04211
Certifying the positivity of trigonometric polynomials is of first importance for design problems in discrete-time signal processing. It is well known from the Riesz-Fej\'ez spectral factorization theorem that any trigonometric univariate polynomial
Externí odkaz:
http://arxiv.org/abs/2202.06544
Autor:
Sawall, David, Schweighofer, Markus
Publikováno v:
In Indagationes Mathematicae January 2024 35(1):37-59
Autor:
Schweighofer, Markus
Let $p$ be a real zero polynomial in $n$ variables. Then $p$ defines a rigidly convex set $C(p)$. We construct a linear matrix inequality of size $n+1$ in the same $n$ variables that depends only on the cubic part of $p$ and defines a spectrahedron $
Externí odkaz:
http://arxiv.org/abs/1907.13611
A quadrature rule of a measure $\mu$ on the real line represents a convex combination of finitely many evaluations at points, called nodes, that agrees with integration against $\mu$ for all polynomials up to some fixed degree. In this paper, we pres
Externí odkaz:
http://arxiv.org/abs/1805.12047
Autor:
Kriel, Tom-Lukas, Schweighofer, Markus
Consider a finite system of non-strict polynomial inequalities with solution set $S\subseteq\mathbb R^n$. Its Lasserre relaxation of degree $d$ is a certain natural linear matrix inequality in the original variables and one additional variable for ea
Externí odkaz:
http://arxiv.org/abs/1710.07521
It is well-known that every non-negative univariate real polynomial can be written as the sum of two polynomial squares with real coefficients. When one allows a weighted sum of finitely many squares instead of a sum of two squares, then one can choo
Externí odkaz:
http://arxiv.org/abs/1706.03941
Autor:
Schweighofer, Markus, Kriel, Tom-Lukas
Consider a finite system of non-strict real polynomial inequalities and suppose its solution set $S\subseteq\mathbb R^n$ is convex, has nonempty interior and is compact. Suppose that the system satisfies the Archimedean condition, which is slightly s
Externí odkaz:
http://arxiv.org/abs/1704.07231