---
title: Turyn Polynomials and the Quarter-Shift
url: https://www.emergentmind.com/topics/turyn-polynomials
type: topic
---

# Turyn Polynomials and the Quarter-Shift

Searching arXiv for the cited Turyn-polynomial papers and closely related terminology.
Turyn polynomials are most explicitly realized as shifted quadratic-character polynomials obtained from the Legendre symbol, and in contemporary analytic number theory they are treated both as special cases of incomplete mixed character sums and as near-Littlewood polynomials whose extremal behavior is governed by the choice of shift. For an odd prime \(p\), a standard form is
\[
F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),
\]
where \(\left(\frac{\cdot}{p}\right)\) is the Legendre symbol and \(a/p\to\alpha\) is the normalized shift. In the broader framework of shifted character-sum polynomials, Turyn polynomials correspond to the case \(\beta=1\) in
\[
S(\chi,\alpha,\beta,\theta)=\sum_{\alpha q<n<(\alpha+\beta)q}\chi(n)e(n\theta),
\]
so they sit naturally beside Fekete polynomials and other quadratic-character Littlewood-type constructions. Recent work establishes that, for every integer \(k\ge 2\), the asymptotic normalized \(L_{2k}\)-norm profile of Turyn polynomials is minimized at the quarter-shift \(\alpha=\tfrac14\), proving a conjecture of G\"unther and Schmidt [2510.06161]. Closely related work on Mahler’s problem shows that the same quarter-shift regime yields normalized Mahler measure exceeding \(0.95\) after the standard one-coefficient correction that produces genuine \(\pm1\)-coefficient Littlewood polynomials [2405.08281].

## 1. Definition and placement within character-sum polynomial families

In the notation used for incomplete mixed character sums, the central object is
\[
S(\chi,\alpha,\beta,\theta)=\sum_{\alpha q<n<(\alpha+\beta)q}\chi(n)e(n\theta),
\]
for a primitive Dirichlet character \(\chi \bmod q\), where \(e(x):=e^{2\pi i x}\). This can be viewed as the polynomial
\[
\sum_{\alpha q<n<(\alpha+\beta)q}\chi(n)x^n
\]
evaluated on the unit circle, and it “naturally generaliz[es] the well-known variants of Fekete (\(\alpha=0\) and \(\beta=1\)) and Turyn (\(\beta=1\)) polynomials.” In this sense, Turyn polynomials are the \(\beta=1\) subfamily of shifted mixed character-sum polynomials [2510.06161].

For quadratic characters, the relevant specialization is
\[
S(\chi_q,\alpha,1,\theta)=\sum_{\alpha q<n<(\alpha+1)q}\chi_q(n)e(n\theta),
\]
where \(\chi_q\) is the quadratic character modulo the prime \(q\). The paper devoted to Mahler’s problem uses an equivalent cyclic-shift formulation based on the Fekete polynomial
\[
F_p(x)=\sum_{j=0}^{p-1}\left(\frac{j}{p}\right)x^j
\]
and defines the Turyn polynomial by
\[
F_{p,t}(x)=\sum_{j=0}^{p-1}\left(\frac{j+t}{p}\right)x^j,
\]
with asymptotic shift parameter \(\alpha=t/p\in[0,1]\) [2405.08281].

This family is therefore best understood as a shifted Legendre-symbol family. In the formulation
\[
F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),
\]
the parameter \(a\) shifts the quadratic-character coefficients, and the normalized asymptotic location is \(\alpha=a/p\). The coefficients are essentially \(\pm1\), with the usual single zero arising from the Legendre symbol at the residue class \(0\); after excluding or correcting that coefficient in the standard way, these become Littlewood polynomials in the usual sense [2510.06161].

A particularly important special case is the quarter-shift \(\alpha=\tfrac14\). The modern literature repeatedly singles this out, both in norm-minimization results and in Mahler-measure computations, as the distinguished Turyn regime [2510.06161].

## 2. Classical form, coefficient structure, and Littlewood-polynomial status

The explicit classical shifted Turyn polynomial appearing in recent analytic work is
\[
F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt).
\]
Here the Legendre symbol furnishes a quadratic-character coefficient sequence. Since \(\left(\frac{0}{p}\right)=0\), the unshifted Fekete polynomial has one zero coefficient and all other coefficients are \(\pm1\); the same phenomenon persists under cyclic shift for \(F_{p,t}\), where exactly one coefficient is \(0\), namely at the index where \(j+t\equiv 0\pmod p\) [2405.08281].

To obtain genuine Littlewood polynomials, one introduces the companion polynomials
\[
F_{p,t}^{\pm}(x)=F_{p,t}(x)\pm x^{p-t}.
\]
These have all coefficients equal to \(\pm1\), and the asymptotic normalized \(L_q\)-norms and Mahler measure are unchanged by this one-term modification in the sense made precise by the asymptotic results [2405.08281].

The terminology “Turyn polynomial” is therefore exact in one well-defined sense: it denotes a cyclically shifted Fekete polynomial, equivalently a shifted quadratic-character polynomial with \(\beta=1\). At the same time, the broader Turyn literature contains adjacent but distinct objects. Turyn-type sequences are quadruples \((A;B;C;D)\) whose associated sequence polynomials satisfy
\[
A(x)A(x^{-1})+B(x)B(x^{-1})+2C(x)C(x^{-1})+2D(x)D(x^{-1})=6n-2,
\]
which is a different construction, although closely related in autocorrelation theory and Hadamard-matrix applications [1206.4107].

A separate source of terminological ambiguity arises from Turán expressions, defined for a polynomial sequence \(P=\{P_k\}\) by
\[
\mathscr{J}_k(P;x)=P_{k+1}(x)^2-P_{k+2}(x)P_k(x).
\]
These belong to the theory of Turán inequalities and weak Hurwitz stability and are unrelated to Turyn polynomials except for the similarity of names [1405.1638].

## 3. Asymptotic \(L_{2k}\)-norm theory and the quarter-shift theorem

The central asymptotic extremal theorem for Turyn polynomials concerns their normalized even norms. The \(L_\lambda\)-norm convention is
\[
\|P\|_\lambda=\left(\int_0^1 |P(e(t))|^\lambda\,dt\right)^{1/\lambda}.
\]
For Turyn polynomials, G\"unther and Schmidt established the existence of a limiting profile \(\phi_k(\alpha)\) via
\[
\lim_{p\to\infty}\frac{1}{\sqrt p}\|F_{p,a}\|_{2k}=\phi_k(\alpha)
\qquad\text{when } a/p\to\alpha.
\]
They verified for \(k=2,3,4\) that \(\phi_k(\alpha)\) is minimized at \(\alpha=\tfrac14\), and conjectured this for all \(k\ge 5\). The recent resolution is the theorem
\[
\forall k\ge 2,\qquad \argmin_{\alpha\in[0,1]}\phi_k(\alpha)=\frac14,
\]
which proves the conjecture in full [2510.06161].

The statement is asymptotic rather than finite-\(p\). It identifies the minimizing shift of the limiting profile
\[
\phi_k(\alpha)=\lim_{p\to\infty}\frac1{\sqrt p}\|F_{p,a}\|_{2k}
\quad\text{when } a/p\to\alpha,
\]
but does not assert that for every sufficiently large prime \(p\) the minimizing discrete shift must be exactly \(a=\lfloor p/4\rfloor\) or any related finite statement. The contribution is the exact asymptotic minimizer, not an explicit closed-form value of \(\phi_k(1/4)\) [2510.06161].

The proof replaces the recursive combinatorics of G\"unther–Schmidt with a probabilistic representation derived from a quadratic random-process limit. For fixed \(\alpha\) and \(\beta\), the random process \(G_{q,\alpha,\beta}\) converges in distribution to
\[
G_{\alpha,\beta}(t)=\frac{e(\alpha t)}{2 \pi i}\sum_{l\in\mathbb Z}
\frac{e(\alpha l)(e(\beta(l+t))-1)}{l+t}Y(l),
\]
where the \(Y(l)\) are independent \(\{-1,1\}\)-valued random variables. In the Turyn case \(\beta=1\), this yields a probabilistic model for the shifted quadratic polynomial [2510.06161].

The limiting norm profile \(\phi_k(\alpha)\) is then rewritten as an expectation over a random Fourier-type series, and a moment expansion shows that only pairings survive because the independence of the \(Y(m)\) annihilates mixed terms unless indices occur in pairs. This reduces the minimization to the building block
\[
M_{2r,t}(\alpha)
=
\left|\sum_{m\in\mathbb Z}\frac{e(2m\alpha)}{(m+t)^2}\right|^{2r}.
\]
Using an integral representation derived from the Lerch transcendent, the analysis shows that this is minimized at \(2\alpha=\tfrac12\), hence at \(\alpha=\tfrac14\), which propagates back to the full theorem [2510.06161].

The quarter-shift theorem is therefore not merely a computational observation. It is an asymptotic extremal result valid for every even norm \(L_{2k}\) with \(k\ge2\), and it supplies a conceptual explanation for the recurrent appearance of the quarter-shift in Turyn-polynomial flatness phenomena.

## 4. Flatness, Mahler measure, and record behavior near \(\alpha=\tfrac14\)

The Mahler-measure viewpoint places Turyn polynomials inside Mahler’s problem for Littlewood polynomials. For a polynomial
\[
f(x)=\sum_{k=0}^n a_kx^k=a_n\prod_{k=1}^n(x-\beta_k),
\]
the Mahler measure is defined by
\[
\log M(f)=\int_0^1\log|f(e(u))|\,du,
\]
equivalently
\[
M(f)=|a_n|\prod_{k=1}^n\max\{1,|\beta_k|\}.
\]
For unimodular polynomials of degree \(n-1\), Parseval gives \(\|f\|_2=\sqrt n\), so the natural normalized quantity is \(M(f)/\|f\|_2\) [2405.08281].

Recent work analyzes Turyn polynomials through the random process
\[
G_{\mathbb X,\alpha}(x):=\sum_{j\in\mathbb Z}
\frac{e(\alpha j)(e(x)-1)}{2\pi i(j+x)}\,\mathbb X(j),
\qquad x\in[0,1],
\]
where \(\{\mathbb X(j)\}_{j\in\mathbb Z}\) are i.i.d. Rademacher random variables. For every \(\alpha\in[0,1]\) and every \(q>0\),
\[
\lim_{p\to\infty}\frac{\|F_{p,\{\!\{\alpha p\}\!\}}\|_q}{\sqrt p}
=
\kappa_q(\alpha),
\]
where
\[
\kappa_q(\alpha):=
\left(\int_0^1\mathbb E\bigl(|G_{\mathbb X,\alpha}(x)|^q\bigr)\,dx\right)^{1/q},
\]
and for Mahler measure,
\[
\lim_{p\to\infty}\frac{M(F_{p,\{\!\{\alpha p\}\!\}})}{\sqrt p}
=
\kappa_0(\alpha),
\]
where
\[
\kappa_0(\alpha):=
\exp\left(\int_0^1 \mathbb E\bigl(\log|G_{\mathbb X,\alpha}(x)|\bigr)\,dx\right)
\]
[2405.08281].

The numerical study reports that the data suggest \(\kappa_0(\alpha)\) is increasing on \(0\le\alpha\le\tfrac14\), and for the quarter-shift it gives
\[
\lim_{p\to\infty}\frac{M(F_{p,\{\!\{p/4\}\!\}})}{\sqrt p}=0.951\ldots.
\]
This is presented as a numerical estimate rather than a proved closed-form constant. The companion Littlewood polynomials \(F_{p,t}^{\pm}\) have the same asymptotic normalized Mahler measure, so the one-coefficient correction does not disturb the asymptotic value and yields a genuine \(\pm1\)-coefficient family with normalized Mahler measure exceeding \(0.95\) [2405.08281].

The asymptotic \(L_{2k}\)-norm minimization theorem provides an explanatory mechanism for this phenomenon. The quarter-shift Turyn polynomials are described as tending to be “somewhat flat,” and this flatness is presented as a natural reason to expect large Mahler measure. A plausible implication is that the Mahler-measure extremality and the even-norm extremality are manifestations of the same underlying distributional structure, though the Mahler result requires separate logarithmic analysis not used in the \(L_{2k}\)-proof [2510.06161].

## 5. Autocorrelation, Barker sequences, and Turyn-type sequence constructions

Turyn’s name also appears prominently in low-autocorrelation sequence theory, which supplies an adjacent but distinct polynomial context. A binary sequence \(A=(A(1),\dots,A(n))\in\{-1,1\}^n\) has aperiodic autocorrelation
\[
C(u)=\sum_{k=1}^{n-u}A(k)A(k+u),
\qquad 0\le u<n,
\]
and a Barker sequence is one for which
\[
|C(u)|\le 1\qquad (1\le u<n).
\]
Encoding the sequence by the polynomial
\[
A(x)=\sum_{j=0}^{n-1}a_jx^j,\qquad a_j\in\{\pm1\},
\]
the autocorrelations appear in
\[
A(x)A^*(x)=\sum_{u=-(n-1)}^{n-1}C(|u|)\,x^{n-1+u},
\qquad
A^*(x)=x^{n-1}A(x^{-1}).
\]
This makes Barker-sequence theory directly relevant to Turyn-style polynomial questions, even though the resulting objects are not the shifted quadratic-character Turyn polynomials of the analytic number-theory literature [1501.06035].

The odd-length case of Turyn’s conjecture is completely settled: if there exists a Barker sequence of odd length \(n\), then \(n\in\{3,5,7,11,13\}\). The proof in the modern treatment proceeds via run structure, parity, two Turyn–Storer identities, and an autocorrelation-difference contradiction, rather than the original more intricate induction [1501.06035].

A broader flatness-based argument for normalized \(\pm1\)-coefficient Littlewood polynomials asserts that there are no square \(L^2\)-flat sequences
\[
P_q(z)=\frac1{\sqrt q}\sum_{j=0}^{q-1}\epsilon_j z^j,\qquad \epsilon_j\in\{\pm1\},
\]
and derives as a consequence that there are only finitely many Barker sequences. In that framework, square \(L^2\)-flatness is equivalent to \(\|P_q\|_4\to1\), so long Barker-type perfection would force impossible asymptotic flatness [1609.03435].

Another adjacent construction is the theory of Turyn-type sequences \(TT(n)\), quadruples \((A;B;C;D)\) of \(\{\pm1\}\)-sequences whose associated polynomials satisfy the norm identity
\[
N(A)+N(B)+2N(C)+2N(D)=6n-2,
\qquad N(A)=A(x)A(x^{-1}).
\]
This polynomial formulation is explicit, and the paper develops an equivalence relation, a canonical form, enumeration up to \(n\le 32\), and the first example of \(TT(38)\). These objects are not the same as shifted Legendre-symbol Turyn polynomials, but they belong to the same autocorrelation-centered combinatorial ecosystem and share the Littlewood-polynomial perspective [1206.4107].

## 6. Terminological distinctions and conceptual synthesis

The modern literature supports a precise but layered usage of the term “Turyn polynomials.” In analytic number theory, the term refers to shifted Fekete polynomials or, equivalently, shifted quadratic-character polynomials
\[
F_{p,t}(x)=\sum_{j=0}^{p-1}\left(\frac{j+t}{p}\right)x^j,
\]
with normalized shift \(\alpha=t/p\), or to the equivalent form
\[
F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt).
\]
This is the sense in which the quarter-shift theorem and the Mahler-measure record are formulated [2405.08281].

In combinatorial design theory and autocorrelation theory, however, Turyn’s name also labels other constructions: Turyn-type sequences, Turyn–Golay conjectures, and Turyn–Storer identities. These are polynomially encoded through autocorrelation or Laurent-norm identities, but they are structurally distinct from the shifted Legendre-symbol family [1206.4107]. A plausible implication is that the shared nomenclature reflects a common focus on \(\pm1\)-coefficients, flatness, autocorrelation, and extremal behavior rather than a single unified polynomial class.

A further distinction is necessary between Turyn and Turán. Turán expressions, such as
\[
\mathscr{J}_k(P;x)=P_{k+1}(x)^2-P_{k+2}(x)P_k(x),
\]
belong to a different literature on weak Hurwitz stability, Bell polynomials, Chebyshev polynomials, and related zero-location problems. They are not Turyn polynomials, and the similarity of names is merely orthographic [1405.1638].

Taken together, these strands show that Turyn polynomials occupy a central place at the interface of quadratic character sums, Littlewood polynomials, flatness theory, and autocorrelation extremals. Their most definitive modern property is now asymptotic: for every integer \(k\ge2\), the quarter-shift \(\alpha=\tfrac14\) uniquely minimizes the limiting normalized \(L_{2k}\)-norm profile \(\phi_k(\alpha)\) [2510.06161]. Their most prominent current application is to Mahler’s problem, where the same quarter-shift regime produces a genuine Littlewood family with asymptotic normalized Mahler measure about \(0.951\) after the standard one-coefficient correction [2405.08281]. This identifies the quarter-shifted Turyn family as a distinguished extremal locus within the broader theory of quadratic-character Littlewood polynomials.

Source: https://www.emergentmind.com/topics/turyn-polynomials