Waring decompositions of the product of two quadrics: the small rank cases

2025/11/28 by Bhat, Meghana, Carlini, Enrico, Dubey, Saipriya +1
#Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Primary:14N07 #Secondary:13D40

paper · doi:10.48550/arxiv.2511.23035

Abstract

In this paper we study forms of the type (x₁²+ ⋯ +x_m²)(y₁²+ ⋯+y_n²) using projections. For m=1, m=2, and for any n we describe: the forbidden locus, the structure and the Hilbert function of all minimal apolar sets. In particular, we show that every minimal apolar ideal has the same Hilbert function. For m,n ≥ 3, we provide new lower and upper bounds for the Waring rank.

Citations

• Upper bounds for the rank of powers of quadrics
• Geometric complexity theory for product-plus-power
• Secant varieties and the complexity of matrix multiplication
• Real and complex Waring rank of reducible cubic forms
• Power Sum Decompositions of Elementary Symmetric Polynomials
• Symmetric tensors: rank, Strassen's conjecture and e-computability
• Four lectures on secant varieties
• On differences between the border rank and the smoothable rank of a polynomial
• A comparison of different notions of ranks of symmetric tensors
• New lower bounds for the border rank of matrix multiplication
• On the rank of a symmetric form
• Most tensor problems are NP-hard
• Most Tensor Problems Are NP-Hard
• On the ranks and border ranks of symmetric tensors
• On the Gorenstein locus of some punctual Hilbert schemes

Related

• Waring decompositions of special ternary forms with different Hilbert functions
• Decompositions of powers of quadrics
• Waring decompositions of special binomials
• Spaces of Sums of Powers and Real Rank Boundaries
• On the Waring Rank of Binary Forms