The selected answer to How does a plant grow before photosynthesis is possible? indicates that a sprout grows beneath the soil using the food stored within the seed.
Does this limited ready food place a restriction upon the depth to which a seed may be placed and still develop into a plant? What depth will be best applicable for a given plant/tree seed? Is there a mathematical relationship known?
The general mathematical relationship requires an allometric equation. There are not as many for seedlings as there are for mature trees (but see the answer by @Richard-smith for a specific example). But lets start with a basic assumption that the emerging seedling will be a very tall cone (tall = high h to r ratio).
The volume of a cone is:
$V = \frac{1}{3} \pi r^2h$
Then we need to know the density of the seedling to figure out how much carbon a seedling of height h would require. Lets say that the density is 1.5 g/cc (excluding water).
Then, assume that the biomass is 1/2 carbon (a fairly reasonable assumption).
To figure out how tall a seed could grow given carbon reserves, it would be necessary to determine the amount of carbon in storage in the seed. Say there were 5 grams. There will be a carbon cost of reallocating this to growth, say 50%. So, for a seed that has 5g of carbon, its total volume might be calculated as:
(5g * 50% growth respiration)/(50% C content of biomass) * 1.5 g/cc = 7.5 cc
assuming that the seedling will have a radius of 0.2 cm, the height could be calculated as:
$h = \frac{V}{(3 \pi * 0.2^2)} = \frac{V}{0.38}$ = 20 cm (for V = 7.5)
Note: this is just a very quick "back of the envelope" calculation with many assumptions. C is 50% of plant biomass is pretty close (the actually number is closer to 48%). The growth respiration cost is a very rough estimate, and will vary with nitrogen content of the tissue, soil temperature, and etc.. The plant will likely need carbon left to build a leaf, and another resource may be limiting.
To answer your specific questions...
Does nutrient availability limit emergence depth?
Yes, the size of the nutrient store in the seed does impose a theoretical limit on the maximum depth at which seed germination and emergence can take place.
What is the best depth for a given species?
The specific depth which gives best germination for any given species is very unlikely to be the maximum depth at which seeds of that species can germinate. This is because other factors also impose limits on germination depth.
Probably the most important other factor is light - most seeds require the perception of red light in order to trigger germination. For most species then, you are unlikely to find seeds germinating and successfully emerging from depths greater than the penetrance of light.
Another important factor in some latitudes is temperature - many seeds (from 26 families) use dormancy as a way of preventing germination in the wrong seasonal conditions. In species where temperature serves as a cue to alleviate dormancy, there will be a maximum depth at which a given seed can detect the temperature fluctuations at the surface. Beyond a certain depth, surface temperature changes will not perceptibly affect local temperature.
If you ever need to find out the best depth of germination for a given species, many can be found on google, or failing that it is trivial to perform a simple experiment.
Is there a mathematical relationship?
There is a general mathematical relationship, an allometric relationship (i.e. relating size to another trait), between seed size and maximum depth of emergence, described by Bond et al. (1999):
where: dmax = maximum depth of emergence
w = seed weight
c = a constant which varies with phylogeny and environmental conditions.
Many species however will not submit to this equation. For example some large-seeded species (incl. coconut, coco-de-mer, avocado), have large seeds because they may need to enable long-distance growth along the ground in order to locate optimum growth conditions. Seeds of these species will therefore not conform to the depth-size relationship above.
Reference
Generally a rule of thumb in gardening is to plant the seed 1-2 times as deep as the seed is big. a pea can go 2 cm, a poppy can go 2mm.
The starch of the seed is converted into sugars, which causes osmotic absorption of water by the new plant cells, and they have about 100-200kpa of upwards turgic pressure maximum, that's equivalent to more than 50 meters of lithospheric pressure.
So it depends on the size of the seed, the main inhibitor is the energy needed to create the hydrostatic pressure in the stem to maintain pressure and cell growth for the given distance.
In dry clay soil which is like concrete, chances are that the plant will not be able to go up, but in sandy soil you can expect some seeds to pierce many meters of sand, unless there is zero light.
