Erdős 699: an effective discriminant bound for selected maximal positions

P699 research · The effectivity question · The symmetric discriminant

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The discriminant supplies an effective bound for some patterns of small-prime maximal positions, including patterns not excluded by counting the positions alone. This is a refinement of the existing inequalities; it does not make every remaining index effective.

Fix an admissible bad triple $(n,i,j)$, put $c=n-j$, and write

$$\binom ni=UV,\qquad S=\{p:p<i\},\qquad b=\frac{i(i-1)}2,\qquad w_t=2i-2-t.$$

For each $p\in S$, choose $r_p\in\{0,\ldots,i-1\}$ maximizing $a_p=v_p(n-r_p)$, and set

$$T=\{r_p:p\in S\},\qquad M_t=\prod_{p:r_p=t}p^{a_p},\qquad W(T)=\sum_{t\in T}(2i-2-t).\tag{1}$$

Thus $M_t\mid n-t$ and $M_t\le n$. Define

$$\kappa_i=\begin{cases}1,&i\text{ composite},\\i^i,&i\text{ prime},\end{cases} \qquad C_i=\frac{\kappa_i\bigl(2(i-1)!\bigr)^{2i-2}}{4^b}.\tag{2}$$

Proposition. If $n\ge i(i-1)$, then

$$n^b\le C_i\prod_{t\in T}M_t^{w_t}.\tag{3}$$

In particular, if $W(T)<b$, then the explicit bound

$$n\le C_i^{1/(b-W(T))}\tag{4}$$

holds. If $|T|\ge2$ and $W(T)=b$, an effective row bound still exists, by the effective $S$-part theorem. Its constants have not been evaluated here.

Proof of the weighted inequality

The orbit polynomial

$$R(z)=\sum_{r=0}^i\binom j{i-r}\binom cr z^r$$

has every coefficient divisible by $V$, so $P=R/V$ is an integer polynomial with nonzero discriminant. The retained symmetric formula is

$$|\operatorname{Disc}R|= \frac{K_i}{(i!)^{2i-2}} \prod_{t=0}^{i-1}\bigl((j-t)(c-t)\bigr)^{i-1-t} \prod_{t=0}^{i-1}(n-t)^t,\qquad K_i=\prod_{r=1}^i r^r.\tag{5}$$

Let $[x]_S$ denote the $S$-part of a positive integer. Since $V$ is coprime to every prime in $S$, formula (5) gives

$$|\operatorname{Disc}P|\ge \frac{[K_i]_S}{[i!]_S^{2i-2}} \prod_{t=0}^{i-1}[n-t]_S^t.\tag{6}$$

This remains valid when the displayed lower bound is not an integer: the valuation inequality holds at every $p\in S$, while the nonzero integer discriminant has nonnegative valuations at all other primes.

Put $\gamma=1-i(i-1)/(2n)$. The bounds

$$\binom ni\ge\frac{\gamma n^i}{i!},\qquad |\operatorname{Disc}R|\le \frac{K_i n^{3b}}{(i!)^{2i-2}4^b}$$

give

$$|\operatorname{Disc}P|\le \frac{K_iU^{2i-2}}{4^b\gamma^{2i-2}n^b}.\tag{7}$$

Combining (6) and (7), and using $U=[\prod_{t=0}^{i-1}(n-t)]_S/[i!]_S$, yields the direct weighted $S$-part inequality

$$4^b\gamma^{2i-2}n^b \le\kappa_i \left[\prod_{t=0}^{i-1}(n-t)^{2i-2-t}\right]_S.\tag{8}$$

Here $K_i/[K_i]_S=\kappa_i$, since every prime factor of $K_i$ is at most $i$.

For $r\ne r_p$, maximality and the ultrametric inequality give $v_p(n-r)\le v_p(r-r_p)$. Consequently

$$\begin{aligned} v_p\!\left(\prod_{r=0}^{i-1}(n-r)^{w_r}\right) &\le w_{r_p}a_p+ \sum_{r\ne r_p}w_rv_p(|r-r_p|)\\ &\le w_{r_p}a_p+(2i-2)v_p((i-1)!). \end{aligned}$$

It follows that the $S$-part in (8) is at most

$$((i-1)!)^{2i-2}\prod_{t\in T}M_t^{w_t}.$$

Finally $n\ge i(i-1)$ gives $\gamma\ge1/2$, proving (3).

For the equality case, use

$$\prod_{t\in T}M_t^{w_t} \le[g_T(n)]_S,\qquad g_T(X)=\prod_{t\in T}(X-t)^{w_t}.$$

The polynomial $g_T$ has at least two distinct roots and degree $W(T)$. Bugeaud--Evertse--Győry's effective Theorem 2.2, in the form already used in the effectivity note, gives effectively computable $A_T,\epsilon_T>0$ with $[g_T(n)]_S\le A_Tn^{W(T)-\epsilon_T}$. This proves an effective cutoff when $W(T)=b$ as well.

What the criterion adds

The positions nearer $i-1$ have smaller weights. For example, at $i=27$ the earlier count bound only handles at most eight distinct maximal positions. This discriminant criterion also handles every nine-position set satisfying

$$\sum_{t\in T}t\ge117,$$

because $9\cdot52-27\cdot26/2=117$. Equality uses the effective $S$-part theorem; a strict inequality gives (4).

For the other two nearby indices whose old exponent deficit is about $1/3$, the analogous sufficient conditions are

$$\begin{array}{c|c|c} i&|T|&\text{sufficient condition}\\\hline 30&10&\sum_{t\in T}t\ge145\\ 33&11&\sum_{t\in T}t\ge176. \end{array}$$

These thresholds are restrictions on finite sets of positions, not a search over rows. Patterns with smaller sums remain to be handled by other arguments. Replacing the integer-discriminant bound by a fixed lower bound for totally real field discriminants can change constants, but by itself does not improve the exponent in (8).