Related papers: Maximum Waring ranks of monomials

In this paper we compute the Waring rank of any polynomial of the form F=M_1+...+M_r, where the M_i are pairwise coprime monomials, i.e., GCD(M_i,M_j)=1 for i not j. In particular, we determine the Waring rank of any monomial. As an…

In this paper we study the complex simultaneous Waring rank for collections of monomials. For general collections we provide a lower bound, whereas for special collections we provide a formula for the simultaneous Waring rank. Our approach…

We describe some forms with greater Waring rank than previous examples. In $3$ variables we give forms of odd degree with strictly greater rank than the ranks of monomials, the previously highest known rank. This narrows the possible range…

In this paper we consider the Waring rank of monomials over the real and the rational numbers. We give a new upper bound for it by establishing a way in which one can take a structured apolar set for any given monomial…

In this paper we study the real rank of monomials and we give an upper bound for the real rank of all monomials. We show that the real and the complex ranks of a monomial coincide if and only if the least exponent is equal to one.

In the polynomial ring $T=k[y_1,...,y_n]$, with $n>1$, we bound the multiplicity of homogeneous radical ideals $I\subset (y_1^{a_1},...,y_n^{a_n})$ such that $T/I$ is a graded $k$-algebra with Krull dimension one. As a consequence we solve…

We show that if a homogeneous polynomial $f$ in $n$ variables has Waring rank $n+1$, then the corresponding projective hypersurface $f=0$ has at most isolated singularities, and the type of these singularities is completely determined by…

We show that for several notions of rank including tensor rank, Waring rank, and generalized rank with respect to a projective variety, the maximum value of rank is at most twice the generic rank. We show that over the real numbers, the…

We give an explicit formula for the Waring rank of every binary binomial form with complex coefficients. We give several examples to illustrate this, and compare the Waring rank and the real Waring rank for binary binomial forms.

Motivated by recent results on the Waring problem for polynomial rings and representation of monomial as sum of powers of linear forms, we consider the problem of presenting monomials of degree kd as sums of k-th powers of forms of degree…

A Waring decomposition of a polynomial is an expression of the polynomial as a sum of powers of linear forms, where the number of summands is minimal possible. We prove that any Waring decomposition of a monomial is obtained from a complete…

A notion of open rank, related with generic power sum decompositions of forms, has recently been introduced in the literature. The main result here is that the maximum open rank for plane quartics is eight. In particular, this gives the…

We prove that if a degree-$d$ homogeneous polynomial $f$ has border Waring rank $\underline{\mathrm{WR}}({f}) = r$, then its Waring rank is bounded by \[ {\mathrm{WR}}({f}) \leq d \cdot r^{O(\sqrt{r})}. \] This result significantly improves…

We give a new differential proof of our result on the maximal rank of generic unions of points of multiplicity two in projective space in degrees greater than five. This simplifies somewhat our proof of the Waring conjecture.

We prove that a general polynomial vector $(f_1, f_2, f_3)$ in three homogeneous variables of degrees $(3,3,4)$ has a unique Waring decomposition of rank 7. This is the first new case we are aware, and likely the last one, after five…

Let F and G be homogeneous polynomials in disjoint sets of variables. We prove that the Waring rank is additive, thus proving the symmetric Strassen conjecture, when either F or G is a power, or F and G have two variables, or either F or G…

We exhibit, for each even degree, a ternary form of rank strictly greater than the maximum rank of monomials. Together with an earlier result in the odd case, this gives the lower bound…

The Waring rank of the generic $d \times d$ determinant is bounded above by $d \cdot d!$. This improves previous upper bounds, which were of the form an exponential times the factorial. Our upper bound comes from an explicit power sum…

We prove a new lower bound for the Mahler measure of a polynomial in one and in several variables that depends on the complex coefficients, and the number of monomials. In one variable our result generalizes a classical inequality of…

The $K$-rank of a binary form $f$ in $K[x,y],~K\subseteq \mathbb{C},$ is the smallest number of $d$-th powers of linear forms over $K$ of which $f$ is a $K$-linear combination. We provide lower bounds for the $\mathbb{C}$-rank (Waring rank)…