More on the sum-freedom of the multiplicative inverse function

Abstract In two papers entitled “Two generalizations of almost perfect nonlinearity” and “On the vector subspaces of $$\mathbb F_{2^n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> </mml:msub> </mml:math> over which the multiplicative inverse function sums to zero”, the first author has introduced and studied the notion of sum-freedom of vectorial functions, which expresses that a function sums to nonzero values over all affine subspaces of $$\mathbb {F}_{2^n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> </mml:msub> </mml:math> of a given dimension $$k\ge 2$$ <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:mrow> </mml:math> , and he then focused on the k th order sum-freedom of the multiplicative inverse function $$x\in \mathbb {F}_{2^n}\mapsto x^{2^n-2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> </mml:msub> <mml:mo>↦</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . Some general results were given for this function (in particular, the case of affine spaces that do not contain 0 was solved positively), and the cases of $$k\in \{3,4,n-4,n-3\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>{</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>3</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> and of k not co-prime with n were solved as well (negatively); but the cases of those linear subspaces of dimension $$k\in \llbracket 5;n-5\rrbracket $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>〚</mml:mo> <mml:mn>5</mml:mn> <mml:mo>;</mml:mo> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>5</mml:mn> <mml:mo>〛</mml:mo> </mml:mrow> </mml:math> , co-prime with n , were left open. The present paper is a continuation of the previous work. After studying, from two different angles, the particular case of those linear subspaces that are stable under the Frobenius automorphism, we deduce from the second approach that, for k small enough (approximately, $$3\le k\le n/10$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>k</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>n</mml:mi> <mml:mo>/</mml:mo> <mml:mn>10</mml:mn> </mml:mrow> </mml:math> ), the multiplicative inverse function is not k th order sum-free. Finally, we deduce from results previously obtained in the second paper mentioned above, that for any even n and every $$2\le k\le n-2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>k</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , the multiplicative inverse function is not k th order sum-free.