Bounds on the higher degree Erdős–Ginzburg–Ziv constants over $${\mathbb {F}}_q^n$$

Abstract The classical Erdős–Ginzburg–Ziv constant of a group G denotes the smallest positive integer $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> such that any sequence S of length at least $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> contains a zero-sum subsequence of length $$\exp (G).$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>exp</mml:mo> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> In the recent paper (Integers 22: Paper No. A102, 17 pp., 2022), Caro and Schmitt generalized this concept, using the m -th degree symmetric polynomial $$e_m(S)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>e</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> instead of the sum of the elements of S and considering subsequences of a given length t . In particular, they defined the higher degree Erdős–Ginzburg–Ziv constants EGZ ( t , R , m ) of a finite commutative ring R and presented several lower and upper bounds to these constants. This paper aims to provide lower and upper bounds for EGZ ( t , R , m ) in case $$R={\mathbb {F}}_q^{n}.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> <mml:mi>n</mml:mi> </mml:msubsup> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> The lower bounds here presented have been obtained, respectively, using the Lovász local lemma and the expurgation method and, for sufficiently large n , they beat the lower bound provided by Caro and Schmitt for the same kind of rings. Finally, we prove closed form upper bounds derived from the Ellenberg–Gijswijt and Sauermann results for the cap-set problem assuming that $$q = p^k,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>p</mml:mi> <mml:mi>k</mml:mi> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> $$t = p,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> and $$m=p-1.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mi>p</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> Moreover, using the slice rank method, we derive a convex optimization problem that provides the best bounds for $$q = 3^k,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mn>3</mml:mn> <mml:mi>k</mml:mi> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> $$t = 3,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> $$m=2,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> and $$k=2, 3,4,5.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mn>5</mml:mn> <mml:mo>.</mml:mo> </mml:mrow> </mml:math>