We use the Hilbertʼs Nullstellensatz (Hilbertʼs Zero Point Theorem) to give a direct proof of the formula for the determinants of the products of tensors. By using this determinant formula and using ...tensor product to represent the transformations of the slices of tensors, we prove some basic properties of the determinants of tensors which are the generalizations of the corresponding properties of the determinants for matrices. We also study the determinants of tensors after two types of transposes. We use the permutational similarity of tensors to discuss the relation between weakly reducible tensors and the triangular block tensors, and give a canonical form of the weakly reducible tensors.
We prove two variations of Hilbert's Nullstellensatz in the univariate case:
(1) If h and
are polynomials in
and there is a positive integer m and a positive constant C such that
then there are n ...polynomials
such that
with the degree bound
(2) If
and h are polynomials in
and there is a positive integer m and a positive constant C such that
then there are n polynomials
such that
with the degree bound
This paper proves such a new Hilbert’s Nullstellensatz for analytic trigonometric polynomials that if {fj}j=1n≥2 are analytic trigonometric polynomials without common zero in the finite complex plane ...ℂ then there are analytic trigonometric polynomials {gj}j=1n≥2 obeying ∑j=1n≥2fjgj=1 in ℂ, thereby not only strengthening Helmer’s Principal Ideal Theorem for entire functions, but also finding an intrinsic path from Hilbert’s Nullstellensatz for analytic polynomials to Pythagoras’ Identity on ℂ.
We develop the foundations of a theory of algebraic geometry for semirings, concentrating mainly on the semiring of tropical polynomials. Replacing ideals with the more general notion of congruences, ...we establish a relationship between congruences on the semiring of tropical polynomials and subsets of tropical space. We ultimately establish analogues of the weak and strong forms of Hilbert's Nullstellensatz.
In this note we describe how Lasoń's generalization of Alon's Combinatorial Nullstellensatz gives a framework for constructing lower bounds on the Turán number ex(n,Ks1,…,sr(r)) of the complete ...r-partite r-uniform hypergraph Ks1,…,sr(r). To illustrate the potential of this method, we give a short and simple explicit construction for the Erdős box problem, showing that ex(n,K2,…,2(r))=Ω(nr−1/r), which asymptotically matches best known bounds when r≤4.
Let I be a proper left ideal in the ring Hx1,…,xn of polynomials in n central variables over the quaternion algebra H. Then there exists a point a=(a1,…,an)∈Hn with aiaj=ajai for all i,j, such that ...every polynomial in I vanishes at a. This generalizes a theorem of Jacobson, who proved the case n=1. Moreover, a polynomial f∈Hx1,…,xn vanishes at all common zeroes of polynomials in I if and only if f belongs to the intersection of all completely prime left ideals that contain I – a notion introduced by Reyes in 2010.
Let G be a simple graph. Assume that c is a mapping from the edges of G to the set {1,2,…,k}. We say that c is neighbor sum distinguishing if every pair of vertex sums of adjacent vertices are ...pairwise distinct, where the vertex sum of a vertex is the sum of the colors on the edges incident to it. The least number k constructing such a coloring is denoted by χΣ′(G). We investigate the challenging conjecture presuming that χΣ′(G)≤Δ(G)+2 for any connected graph G≠C5, which is proposed by Flandrin et al. in 2012. We support this conjecture by proving that χΣ′(G)≤Δ(G)+⌈5col(G)+12⌉, which improves the previous bound Δ(G)+3col(G)−4 proved in Przybyło and Wong, 2015. Furthermore, our result holds for the list version.
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