\documentclass[11pt]{amsart}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{booktabs,array,longtable}
\usepackage{microtype}
\usepackage{needspace}
\usepackage[hidelinks]{hyperref}
\hypersetup{pdftitle={The L'vov-Kaplansky Conjecture for Standard Polynomials},pdfauthor={Y. T. Miah},pdfsubject={Set-of-values classification for alternating standard polynomials}}
\usepackage{enumitem}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\Mat}{\mathrm{M}}
\newcommand{\im}{\operatorname{Im}}
\newcommand{\tr}{\operatorname{tr}}
\newcommand{\diag}{\operatorname{diag}}
\newcommand{\Span}{\operatorname{span}}
\newcommand{\rk}{\operatorname{rank}}
\newcommand{\one}{\mathbf{1}}
\newcommand{\ord}{\operatorname{ord}}
\newcommand{\Sl}{\mathfrak{sl}}
\newcommand{\Gam}{\Gamma}
\title[L'vov--Kaplansky for Standard Polynomials]{The L'vov--Kaplansky Conjecture for Standard Polynomials}
\author{Y. T. Miah}
\date{7 November 2026}
\subjclass[2020]{16R10, 15A24, 16R30}
\keywords{L'vov--Kaplansky conjecture, standard polynomial, matrix polynomial image, alternating polynomial, Euler trail, generic division algebra}
\emergencystretch=2em
\begin{document}
\begin{abstract}
The L'vov--Kaplansky conjecture asks whether the image of a multilinear
polynomial on a full matrix algebra is a vector subspace. For the
unnormalized standard polynomial $s_d$ over an algebraically closed field
$K$ of characteristic zero, the sets of individual matrix values are
determined in every degree and order. The image of $s_{2r-1}$ is zero
when $n<r$ and all of $\Mat_n(K)$ when $n\ge r$; the image of $s_{2r}$
is zero when $n\le r$ and $\Sl_n(K)$ when $n\ge r+1$.
The odd-degree proof uses explicit linear slices and Jordan forms.
The even-degree argument combines a uniform nilpotent construction and
an additive transfer for admissible cores with generic-nullity results
of Brassil--Reichstein and Lin--Tang--Zhao, extending a method of
Qiyue Tang to obtain actual polynomial values.
\end{abstract}
\maketitle

\section{Introduction and the classification}\label{sec:introduction}
The L'vov--Kaplansky conjecture concerns the images of multilinear
noncommutative polynomials evaluated on full matrix algebras. In the
setting of algebraically closed fields of characteristic zero, it asks
whether each such image is a vector subspace
\cite{KanelBelovMalevRowen2016,KanelBelovEtAl2020}.
The alternating standard polynomials constitute a distinguished family
for which one can go further and identify the image explicitly, in all
degrees and matrix orders.

For every integer $d\ge1$ define the \emph{unnormalized standard polynomial}
\begin{equation}\label{eq:standard}
 s_d(X_1,\ldots,X_d)=\sum_{\sigma\in S_d}\operatorname{sgn}(\sigma)
 X_{\sigma(1)}\cdots X_{\sigma(d)}.
\end{equation}
The image $\im s_d(\Mat_n(K))$ means the set of matrices attained as
$s_d(A_1,\ldots,A_d)$ for \emph{one tuple} of inputs; it is not the
linear span or Zariski closure of those values. Write
$\Sl_n(K)=\{T\in\Mat_n(K):\tr T=0\}$. Matrix units are indexed from
zero except when quoting another author's conventions. Throughout the
article, $K$ is algebraically closed of characteristic zero unless a
statement explicitly declares otherwise.

\Needspace{13\baselineskip}
\begin{theorem}[Images of the standard polynomials]\label{thm:main}
For every algebraically closed field $K$ of characteristic zero and all
integers $r,n\ge1$, the \emph{actual images} are
\begin{equation}\label{eq:main}
\begin{aligned}
 \im s_{2r-1}(\Mat_n(K))
 &=\begin{cases}\{0\},&n<r,\\ \Mat_n(K),&n\ge r,\end{cases}\\[3pt]
 \im s_{2r}(\Mat_n(K))
 &=\begin{cases}\{0\},&n\le r,\\ \Sl_n(K),&n\ge r+1.\end{cases}
\end{aligned}
\end{equation}
\end{theorem}

The zero branches follow from the Amitsur--Levitzki identity and
its standard recursion \cite{AmitsurLevitzki1950}. In even degree,
cyclically rotating a word reverses its permutation sign while
preserving its trace, so every value is traceless. The main task is
therefore to attain each matrix in the two positive branches of
\eqref{eq:main}.

The lowest-degree cases are well established. The ordinary commutator
$s_2$ attains every traceless matrix \cite{AlbertMuckenhoupt1957},
and Vitas's cubic image-linearity theorem gives the corresponding
$s_3$ result \cite{Vitas2026}. The quartic result of Fagundes,
de Mello and da Silva dos Santos includes the actual-image assertion
for $s_4$ \cite[Theorem~8, Case~1]{FagundesMelloSantos2022}.
Qiyue Tang established the degree-five and degree-six conclusions,
including the actual traceless image of $s_6$ in orders at least four
\cite[Appendix~A, Proposition~11]{Tang2026FiveSix}.

Classical antecedents include Kostant's work on standard identities,
Swan's graph method and its correction, and results of Rowen and
Hutchinson on standard polynomials and matrix identities
\cite{Kostant1958,Swan1963,Swan1969Correction,Rowen1974,Hutchinson1974}.
Serre studied special matrix values and related norm questions
\cite{Serre2016}. These works provide context for the exact
all-degree image classification undertaken here.

The two parity arguments have different structures. Part~I establishes
the odd-degree assertion through signed path and cycle evaluations,
fixed linear slices and target-dependent Jordan constructions at the
minimal size. Part~II uses a rank-one nilpotent with surjective full
differential to obtain actual values. Its nilpotent and
projective-incidence method builds on Tang's Appendix~A, while the
uniform seed, additive-core transfer and exact degree-eight and
degree-ten cases supply further constructions. The general even-degree
step uses the generic-nullity theorem of Brassil--Reichstein
\cite[Theorem~1.2]{BrassilReichstein2022} and Chen Lin, Chenhao Tang
and Enhan Zhao \cite[Theorem~1.2]{LinTangZhao2026}. The final
appendices give the exact finite checks supporting specified steps.

\part*{I. Odd standard polynomials}
\input{odd_proof}
\part*{II. Even standard polynomials}
\input{even_proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
The odd assertion is proved in Section~\ref{sec:assembly}. The even
assertion follows from Theorem~\ref{thm:even-complete}, including the
identity and low-order cases. Both arguments establish equality of
sets of individual values: by explicit slices in odd degree and by
the attained-nilpotent criterion in even degree.
\end{proof}

\section*{Use of AI}
OpenAI's GPT-6 Pro assisted substantially in the mathematical
investigation, development of proofs, exact computation, source
assessment and manuscript preparation. The author is responsible for
the mathematical claims, attribution and final presentation.

\appendix
\input{verification}
\bibliographystyle{amsplain}
\urlstyle{same}
{\raggedright\interlinepenalty=10000
\bibliography{references}
}
\end{document}