Representations of non-negative polynomials having finitely many zeros

Murray Marshall

Displaying similar documents to “Representations of non-negative polynomials having finitely many zeros”

On a generalization of de Rham lemma

Kyoji Saito (1976)

Annales de l'institut Fourier

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Let $M$ be a free module over a noetherian ring. For $ω_{1}, ..., ω_{k} \in M$ , let $𝒜$ be the ideal generated by coefficients of $ω_{1} \land ... \land ω_{k}$ . For an element $ω \in ⋀^{p} M$ with $p < prof . 𝒜$ , if $ω \land ω_{1} \land ... \land ω_{k} = 0$ , there exists $η_{1}, ..., η_{k} \in ⋀^{p - 1} M$ such that $ω = \sum_{i = 1}^{k} η_{i} \land ω_{i}$ . This is a generalization of a lemma on the division of forms due to de Rham (Comment. Math. Helv., 28 (1954)) and has some applications to the study of singularities.

An elementary proof of the Briançon-Skoda theorem

Jacob Sznajdman (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

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We give an elementary proof of the Briançon-Skoda theorem. The theorem gives a criterionfor when a function $φ$ belongs to an ideal $I$ of the ring of germs of analytic functions at $0 \in ℂ^{n}$ ; more precisely, the ideal membership is obtained if a function associated with $φ$ and $I$ is locally square integrable. If $I$ can be generated by $m$ elements,it follows in particular that $\bar{I^{min (m, n)}} \subset I$ , where $\bar{J}$ denotes the integral closure of an ideal $J$ .

On prime values of reducible quadratic polynomials

W. Narkiewicz, T. Pezda (2002)

Colloquium Mathematicae

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It is shown that Dickson’s Conjecture about primes in linear polynomials implies that if f is a reducible quadratic polynomial with integral coefficients and non-zero discriminant then for every r there exists an integer $N_{r}$ such that the polynomial $f (X) / N_{r}$ represents at least r distinct primes.

Relationship of certain rings of infinite matrices over integers

Mario Petrich, Pedro V. Silva (2000)

Bollettino dell'Unione Matematica Italiana

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Sia $N$ l'insieme degli interi non negativi e $Z$ l'anello degli interi. Sia $A$ l'anello delle matrici $N \times N$ su $Z$ che hanno solo un numero finito di cifre non nulle in ogni linea ed in ogni colonna. Sia $B$ il sottoanello generato da $X$ e $Y$ , dove $X$ (rispettivamente $Y$ ) è ottenuto dalla matrice identità muovendo gli 1 una posizione a destra (rispettivamente in giù). Sia pure $C$ il sottoanello di $A$ generato da $1 - X$ e $1 - Y$ . Infine sia $F$ il sottoanello delle matrici di $A$ che hanno solo un numero finito di cifre...

Extension of the Two-Variable Pierce-Birkhoff conjecture to generalized polynomials

Charles N. Delzell (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

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Let $h : ℝ^{n} \to ℝ$ be a continuous, piecewise-polynomial function. The Pierce-Birkhoff conjecture (1956) is that any such $h$ is representable in the form ${sup}_{i} {inf}_{j} f_{i j}$ , for some finite collection of polynomials $f_{i j} \in ℝ [x_{1}, ..., x_{n}]$ . (A simple example is $h (x_{1}) = | x_{1} | = sup {x_{1}, - x_{1}}$ .) In 1984, L. Mahé and, independently, G. Efroymson, proved this for $n \leq 2$ ; it remains open for $n \geq 3$ . In this paper we prove an analogous result for “generalized polynomials” (also known as signomials), i.e., where the exponents are allowed to be arbitrary real numbers, and not just...