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We generalize an example, due to Sylvester, and prove that any monomial of degree $d$ in $mathbb R[x_0, x_1]$, which is not a power of a variable, cannot be written as a linear combination of fewer than $d$ powers of linear forms.
For $2$ vectors $x,yin mathbb{R}^m$, we use the notation $x * y =(x_1y_1,ldots ,x_my_m)$, and if $x=y$ we also use the notation $x^2=x*x$ and define by induction $x^k=x*(x^{k-1})$. We use $<,>$ for the usual inner product on $mathbb{R}^m$. For $A$ an
This paper develops our previous work on properness of a class of maps related to the Jacobian conjecture. The paper has two main parts: - In part 1, we explore properties of the set of non-proper values $S_f$ (as introduced by Z. Jelonek) of these
In [9], Migliore, Miro-Roig and Nagel, proved that if $R = mathbb{K}[x,y,z]$, where $mathbb{K}$ is a field of characteristic zero, and $I=(L_1^{a_1},dots,L_r^{a_4})$ is an ideal generated by powers of 4 general linear forms, then the multiplication b
Parking functions are a widely studied class of combinatorial objects, with connections to several branches of mathematics. On the algebraic side, parking functions can be identified with the standard monomials of $M_n$, a certain monomial ideal in t
We determine the Waring ranks of all sextic binary forms using a Geometric Invariant Theory approach. In particular, we shed new light on a claim by E. B. Elliott at the end of the 19th century concerning the binary sextics with Waring rank 3.