Sum of squares length of real forms

Abstract

Abstract. For n, d ≥ 1 let p(n, 2d) denote the smallest number p such that every sum of squares of forms of degree d in R[x 1 , . . . , xn] is a sum of p squares. We establish lower bounds for these numbers that are considerably stronger than the bounds known so far. Combined with known upper bounds they give p(3, 2d) ∈ {d + 1, d + 2} in the ternary case. Assuming a conjecture of Iarrobino-Kanev on dimensions of tangent spaces to catalecticant varieties, we show that p(n, 2d) ∼ const · d (n−1)/2 for d → ∞ and all n ≥ 3. For ternary sextics and quaternary quartics we determine the exact value of the invariant, showing p(3, 6) = 4 and p(4, 4) = 5.

Approximate Pythagoras Numbers on $*$-algebras over $\mathbb{C}$

“…and in [6] it is proven (up to a conjecture of Iarrobino-Kanev from algebraic geometry) that this is asymptotically tight as d → ∞.…”

“…This bound is tight for n 2 but it is still unclear whether it can be improved for n 3. The only known lower bound on this number is n+2 (see for example [6] for a nice overview).…”

“…forms) of degree 2d. We do not state the known results in detail, but refer to [6] instead. For a few small cases of n and d the value is known exactly, and in the general case an upper bound is known.…”

Sums of Squares: A Real Projective Story

“…For all positive integers and , what is py( (ℙ ))? To date, the strongest results are due to Claus Scheiderer [24]. He proves that py( (ℙ 2 )) ∈ { + 1, + 2}.…”

Sums of Squares and Quadratic Persistence on Real Projective Varieties

“…Unlike the intervening work on rational functions, [CLR95] again concentrates on homogeneous polynomials, providing both lower and upper bounds on their Pythagoras numbers. Advancing these ideas, [Sch17] establishes new lower bounds that are much closer to the existing upper bounds. By reproving Theorem 3.6 in [Sch17], Example 5.11 hints at the universality of our geometric paradigm.…”