D-Optimality for regression of polynomial models in one variable with missing terms Let's say I have a model that looks as follows:
$$y = x + ax^3 + bx^5 + cx^7 + dx^9$$
Given $n$ free choices for x as input measurements how can I determine which $x$'s I should input to best determine $a,b,c,d$ coefficients. The range for $x$ is $(0,t)$.
 A: In your polynomial model there is no coefficient for the linear term and no intercept (and no error term ...) I will assume that is an oversight. The classic book Optimal Design of Experiments by  Frederick Pukelsheim analyzes D-optimal designs for polynomial regression, but for models including all polynomial terms up to $d$ ($x^d$). Let us see how this works for your model.
We will use, as in that book, $x$ normalized to the range $[-1, 1]$. As D-optimality is invariant under linear transformations of the predictors, you can just transform the results back to your range. Then, there is the approximate result (for large $d$) that the support points for a D-optimal design can be got from the arcsine distribution:
$$ x_i = \cos\left(\frac{d-i}{d}\pi\right), \quad i=0,1,\dotsc,d
$$
Let us use the package AlgDesign in R to check this:
library(AlgDesign)  
d <- 9
A <- data.frame(x=cos(pi*(d-0:d)/d))  
a.des <- optFederov(~ 1 + x + I(x^3) + I(x^5) + I(x^7) + I(x^9), A,
                    approximate=TRUE, evaluateI=TRUE)

a.des
$D
[1] 0.006375102

$A
[1] 115076.8

$I
[1] 6

$Ge
[1] 1

$Dea
[1] 1

$design
   Proportion          x
1         0.1 -1.0000000
2         0.1 -0.9396926
3         0.1 -0.7660444
4         0.1 -0.5000000
5         0.1 -0.1736482
6         0.1  0.1736482
7         0.1  0.5000000
8         0.1  0.7660444
9         0.1  0.9396926
10        0.1  1.0000000

So this gives design proportions which are actually all equal, so can be actually realized in an exact design with $N=10$ (or some multiple). You can yourself try this, but replace A with a linearly spaced set, and results will not be so nice.
Let us then try to find an exact design with this support points and $N=20$:
e.des <- optFederov(~ 1 + x + I(x^3) + I(x^5) + I(x^7) + I(x^9), rbind(A, A),
                    approximate=FALSE, evaluateI=TRUE, nTrials=20) 

e.des
$D
[1] 0.006375102

$A
[1] 115074.4

$I
[1] 6

$Ge
[1] 1

$Dea
[1] 1

$design
.
.
.

(the design just consists of the support points replicated twice). Note that Ge, the efficiency compared to the approximate theory design, is 1.
