Minimal Value Set Polynomials (MVSPs)

• MVSPs are nonconstant polynomials over finite fields that achieve the minimal possible value set size determined by their degree.
• They arise at the intersection of finite-field arithmetic, additive and linearized polynomials, and algebraic geometry, aiding in explicit classification and construction.
• Current research employs differential-functional identities and additive structures to classify MVSPs and link them to curve constructions and Frobenius nonclassicality.

Searching arXiv for recent and foundational papers on Minimal Value Set Polynomials and related value-set theory. First, I’ll look for the recent MVSP classification paper and foundational characterization work. Searching arXiv for "Minimal Value Set Polynomials Borges Reis" and related titles. Minimal value set polynomials (MVSPs) are nonconstant polynomials over a finite field whose value sets are as small as the general degree bound allows. For $F\in \mathbb{F}_q[x]$, the value set is $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$, and a basic counting argument gives

$$\left\lfloor \frac{q-1}{\deg F}\right\rfloor+1 \le |V_F| \le q.$$

An MVSP is a polynomial attaining the lower bound; equivalently,

$$|V_F|=\left\lfloor \frac{q-1}{\deg F}\right\rfloor+1=\left\lceil \frac{q}{\deg F}\right\rceil.$$

The subject sits at the intersection of finite-field arithmetic, additive and linearized polynomials, value-set theory, and the geometry of curves over finite fields. Classical work of Carlitz, Lewis, Mills, Straus, Gómez-Calderón, Madden, and others established early structure theorems, while more recent results describe which subsets of $\mathbb{F}_q$ can occur as MVSP value sets, classify large classes of MVSPs, and place them within the broader distribution of value-set sizes (Borges et al., 2011, Borges et al., 9 Aug 2025).

1. Definition, normalization, and extremal character

For a nonconstant polynomial $F\in \mathbb{F}_q[x]$, the extremal problem is to minimize $|V_F|$ subject to $\deg F$. The standard normalization is the lower bound

$$|V_F| \ge \left\lfloor \frac{q-1}{\deg F}\right\rfloor+1,$$

and $F$ is an MVSP precisely when equality holds (Borges et al., 2011). The literature also uses the equivalent form $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$0, emphasized in the recent global classification program (Borges et al., 9 Aug 2025).

The trivial cases $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$1 are exceptional. The recent classification work separates them from the genuinely structural regime $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$2: $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$3 occurs exactly for

$$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$4

and arbitrary $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$5-element value sets can be realized by Lagrange interpolation using $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$6 (Borges et al., 9 Aug 2025). The substantive theory therefore concentrates on MVSPs with more than two values.

A recurrent misconception is to identify MVSPs with permutation polynomials of low defect. The two notions are opposite extremes. A permutation polynomial has $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$7, whereas an MVSP has the smallest value set compatible with its degree. This opposition becomes especially sharp in asymptotic and average-value-set results discussed below.

2. Differential characterization and additive structure

A central structural theorem rewrites the MVSP condition as a differential-functional identity. Let $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$8 with $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$9, and define

$$\left\lfloor \frac{q-1}{\deg F}\right\rfloor+1 \le |V_F| \le q.$$0

Then $$\left\lfloor \frac{q-1}{\deg F}\right\rfloor+1 \le |V_F| \le q.$$1 is an MVSP with $$\left\lfloor \frac{q-1}{\deg F}\right\rfloor+1 \le |V_F| \le q.$$2 if and only if there exists $$\left\lfloor \frac{q-1}{\deg F}\right\rfloor+1 \le |V_F| \le q.$$3 such that

$$\left\lfloor \frac{q-1}{\deg F}\right\rfloor+1 \le |V_F| \le q.$$4

Moreover, $$\left\lfloor \frac{q-1}{\deg F}\right\rfloor+1 \le |V_F| \le q.$$5 for some, in fact every, $$\left\lfloor \frac{q-1}{\deg F}\right\rfloor+1 \le |V_F| \le q.$$6 (Borges et al., 2011). This criterion is the main bridge between combinatorial minimality and algebraic structure.

The same paper packages the condition into the space

$$\left\lfloor \frac{q-1}{\deg F}\right\rfloor+1 \le |V_F| \le q.$$7

for $$\left\lfloor \frac{q-1}{\deg F}\right\rfloor+1 \le |V_F| \le q.$$8 separable, monic, of degree $$\left\lfloor \frac{q-1}{\deg F}\right\rfloor+1 \le |V_F| \le q.$$9, and split over $$|V_F|=\left\lfloor \frac{q-1}{\deg F}\right\rfloor+1=\left\lceil \frac{q}{\deg F}\right\rceil.$$0 (Borges et al., 2011). This space contains the constant roots of $$|V_F|=\left\lfloor \frac{q-1}{\deg F}\right\rfloor+1=\left\lceil \frac{q}{\deg F}\right\rceil.$$1 and the nonconstant MVSPs with value set equal to the root set of $$|V_F|=\left\lfloor \frac{q-1}{\deg F}\right\rfloor+1=\left\lceil \frac{q}{\deg F}\right\rceil.$$2.

The differential criterion forces additive structure. A necessary condition for nontrivial $$|V_F|=\left\lfloor \frac{q-1}{\deg F}\right\rfloor+1=\left\lceil \frac{q}{\deg F}\right\rceil.$$3 is the existence of positive integers $$|V_F|=\left\lfloor \frac{q-1}{\deg F}\right\rfloor+1=\left\lceil \frac{q}{\deg F}\right\rceil.$$4 and $$|V_F|=\left\lfloor \frac{q-1}{\deg F}\right\rfloor+1=\left\lceil \frac{q}{\deg F}\right\rceil.$$5 such that $$|V_F|=\left\lfloor \frac{q-1}{\deg F}\right\rfloor+1=\left\lceil \frac{q}{\deg F}\right\rceil.$$6 and

$$|V_F|=\left\lfloor \frac{q-1}{\deg F}\right\rfloor+1=\left\lceil \frac{q}{\deg F}\right\rceil.$$7

is $$|V_F|=\left\lfloor \frac{q-1}{\deg F}\right\rfloor+1=\left\lceil \frac{q}{\deg F}\right\rceil.$$8-additive (Borges et al., 2011). Conversely, additive polynomials generate large classes of MVSPs. If $$|V_F|=\left\lfloor \frac{q-1}{\deg F}\right\rfloor+1=\left\lceil \frac{q}{\deg F}\right\rceil.$$9, $\mathbb{F}_q$0, and

$\mathbb{F}_q$1

is additive and split, then the map $\mathbb{F}_q$2 sends $\mathbb{F}_q$3 into $\mathbb{F}_q$4 (Borges et al., 2011). This additive reduction is one of the organizing principles of modern MVSP theory.

A particularly important case is $\mathbb{F}_q$5 over $\mathbb{F}_q$6. Then

$\mathbb{F}_q$7

which is exactly the class of polynomials over $\mathbb{F}_q$8 that are MVSPs with value set $\mathbb{F}_q$9, together with constants in $F\in \mathbb{F}_q[x]$0 (Borges et al., 2011).

3. Realizable value sets and the current classification picture

The recent large-scale classification program reformulates the subject as a problem about subsets $F\in \mathbb{F}_q[x]$1. For

$F\in \mathbb{F}_q[x]$2

the basic question is when $F\in \mathbb{F}_q[x]$3 (Borges et al., 9 Aug 2025).

The answer for $F\in \mathbb{F}_q[x]$4 is structural and explicit: $F\in \mathbb{F}_q[x]$5 for some $F\in \mathbb{F}_q[x]$6 with $F\in \mathbb{F}_q[x]$7, some $F\in \mathbb{F}_q[x]$8-subspace $F\in \mathbb{F}_q[x]$9 with $|V_F|$0, and some positive integer $|V_F|$1, where

$|V_F|$2

Thus every nontrivial MVSP value set is, up to affine transformation, a power image of a vector subspace (Borges et al., 9 Aug 2025).

Affine subspaces are the first major special case, corresponding to $|V_F|$3. If $|V_F|$4 is an $|V_F|$5-vector space with $|V_F|$6, $|V_F|$7, and $|V_F|$8 is the smallest positive integer such that $|V_F|$9, then with

$\deg F$0

there exists a monic $\deg F$1-linearized polynomial $\deg F$2 such that

$\deg F$3

and

$\deg F$4

This reduces the affine-subspace case to the previously understood subfield case (Borges et al., 9 Aug 2025).

The same paper proposes a conjectural full classification. Roughly, if $\deg F$5 is not a field, then MVSPs with value set $\deg F$6 should be exactly the $\deg F$7-th powers of MVSPs with value set $\deg F$8; if $\deg F$9 is a field, then they should be specific powers of MVSPs with a minimal containing subfield value set (Borges et al., 9 Aug 2025). The conjecture is confirmed by prior results for $$|V_F| \ge \left\lfloor \frac{q-1}{\deg F}\right\rfloor+1,$$0 or $$|V_F| \ge \left\lfloor \frac{q-1}{\deg F}\right\rfloor+1,$$1, and additional instances, including the cases $$|V_F| \ge \left\lfloor \frac{q-1}{\deg F}\right\rfloor+1,$$2 and $$|V_F| \ge \left\lfloor \frac{q-1}{\deg F}\right\rfloor+1,$$3, are proved there (Borges et al., 9 Aug 2025). In particular, Conjecture $$|V_F| \ge \left\lfloor \frac{q-1}{\deg F}\right\rfloor+1,$$4 holds for $$|V_F| \ge \left\lfloor \frac{q-1}{\deg F}\right\rfloor+1,$$5, yielding an explicit classification up to affine equivalence.

4. Construction paradigms and spectral phenomena

The subfield case remains the most completely understood constructive regime. For $$|V_F| \ge \left\lfloor \frac{q-1}{\deg F}\right\rfloor+1,$$6 over $$|V_F| \ge \left\lfloor \frac{q-1}{\deg F}\right\rfloor+1,$$7, the space $$|V_F| \ge \left\lfloor \frac{q-1}{\deg F}\right\rfloor+1,$$8 is an $$|V_F| \ge \left\lfloor \frac{q-1}{\deg F}\right\rfloor+1,$$9-vector space of dimension $F$0, and every element is built from Galois orbits of monomials whose exponents have base-$F$1 digits in $F$2 (Borges et al., 2011). More precisely, if

$F$3

and $F$4, then sums of the form

$F$5

span the full space (Borges et al., 2011). For $F$6 splitting over $F$7, the corresponding space has $F$8-dimension $F$9 (Borges et al., 2011). This gives exact counts of MVSPs with prescribed subfield-like value sets.

These constructions also produce genuinely new examples. In $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$00, the polynomial

$$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$01

lies in $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$02 for $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$03, hence is an MVSP with value set contained in $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$04; the cited analysis verifies that it is not of either class previously considered by Carlitz and Mills (Borges et al., 2011).

A different construction paradigm, not degree-classified in the classical sense, arises from the class $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$05 obtained by modifying a linear permutation at $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$06 points and then adding the identity. Its spectrum satisfies

$$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$07

so intermediate sizes are absent (Işık et al., 2017). The paper determines $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$08, gives the exact spectrum for $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$09, constructs families with $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$10, and produces polynomials avoiding a prescribed multiplicative coset: $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$11 for a subgroup $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$12 of size $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$13 (Işık et al., 2017). Because the class is not organized by degree, its relation to classical MVSPs is indirect; nonetheless it demonstrates how small and highly structured value sets can be engineered by controlled perturbations of simple permutations.

5. Typical value-set size and the exceptional status of MVSPs

MVSPs are extremal, but the generic situation is very different. For a general degree-$$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$14 polynomial, Birch and Swinnerton-Dyer showed

$$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$15

and $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$16 as $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$17 (Matera et al., 2015). Cohen further showed that for fixed $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$18 there is a finite set $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$19 such that any degree-$$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$20 polynomial satisfies $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$21 for some $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$22 (Mullen et al., 2012).

Average-value-set results make the contrast with MVSPs precise. For structured families $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$23 of monic degree-$$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$24 polynomials defined by algebraic conditions on their coefficients, one has

$$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$25

under geometric hypotheses $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$26, with no restriction on the characteristic of $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$27 (Matera et al., 2015). For the special family obtained by fixing $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$28 consecutive leading coefficients, the average sharpens to

$$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$29

for $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$30 and $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$31 (Cesaratto et al., 2013).

From the standpoint of MVSPs, these results imply that typical value sets are of order $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$32, whereas minimal sizes are on the order of $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$33. The papers on averages do not classify MVSPs, but they explicitly frame MVSPs as deep outliers: if a positive proportion of a large structured family had value sets substantially below $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$34, the average would be forced down, contradicting the asymptotic formula (Matera et al., 2015). This suggests that MVSPs are very rare among geometrically generic families.

A related misconception is that fixing several coefficients should move a family toward minimal behavior. The fixed-coefficient average theorem shows the opposite: even after imposing $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$35 consecutive coefficient constraints, the average remains $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$36 up to a bounded error (Cesaratto et al., 2013).

6. Multivariate extensions and support-sensitive bounds

The univariate MVSP notion has no direct multivariate analogue with a single universally accepted lower bound, but multivariate value-set theory provides upper bounds on $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$37 for non-permutation maps that are directly relevant to extremal questions. For a nonconstant polynomial map

$$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$38

if $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$39, then

$$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$40

which specializes for $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$41 to Wan’s bound

$$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$42

for non-permutation polynomials (Mullen et al., 2012). The same paper explicitly situates this alongside the classical MVSP condition

$$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$43

in the univariate case (Mullen et al., 2012).

Degree alone is often crude. A sharper multivariate approach uses the Newton polytope $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$44 of the associated scalar polynomial and the invariant

$$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$45

Then, if $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$46,

$$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$47

which always improves the degree-only bound because $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$48 (Smith, 2013). The example

$$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$49

has $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$50 and $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$51, so the polytope bound is sharp there (Smith, 2013).

This support-sensitive viewpoint was refined further by introducing the integral dilation factor $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$52 from the full degree matrix of the monomial support. The refined bound is

$$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$53

and one has

$$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$54

These results provide an alternate proof of Kosters’ degree bound, an improved Newton polytope-based bound, and an improvement of a degree matrix-based result due to Zan and Cao (Smith, 2015). For MVSP research, the significance is methodological: extremal small-image behavior in several variables is governed not just by degree, but by the detailed combinatorics of the monomial support.

7. Arithmetic geometry and Frobenius nonclassical curves

MVSPs have substantial geometric consequences. A major recent result states that, assuming the structural conjecture described above, an irreducible plane curve

$$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$55

over $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$56 with $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$57 is $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$58-Frobenius nonclassical if and only if

$$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$59

for some divisor $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$60, and $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$61 is an MVSP with value set $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$62 (Borges et al., 9 Aug 2025). Since the conjecture is proved for $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$63, this becomes unconditional in that case and yields a complete characterization of the $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$64-Frobenius nonclassical curves of type $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$65 there (Borges et al., 9 Aug 2025).

An earlier geometric development uses the class

$$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$66

to construct curves

$$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$67

generalizing the Hermitian curve (Borges et al., 2014). The set $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$68 is an $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$69-vector space of dimension $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$70, and explicit generators are built from trace polynomials and Frobenius orbits (Borges et al., 2014). For a distinguished family $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$71 in this construction, one obtains

$$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$72

where $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$73 is the smallest integer $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$74 with $$V_F:=\{F(a):a\in\mathbb{F}_q\}\subseteq \mathbb{F}_q$$75 (Borges et al., 2014). The same paper determines the Weierstrass semigroup at the unique point at infinity and proves that these curves are Castle curves (Borges et al., 2014).

These geometric applications clarify the broader role of MVSPs. They are not merely extremal examples in value-set combinatorics; they are also a source of explicit high-point curves, Frobenius nonclassicality phenomena, and function fields with tightly controlled ramification. In that sense, MVSPs link finite-field polynomial theory to algebraic geometry in a particularly rigid and productive way.